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arXiv:1701.00026v1 [math.RA] 30 Dec 2016
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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ABYZOV A. N.
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Abstract. We describe rings over which every right module is almost injective. We give a description of rings over which every simple module is a almost projective.
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Let M, N be right R-modules. A module M is called almost N- injective, if for any submodule N of N and any homomorphism f : N M, either there exists a homomorphism g : N M such that f = g or there exists a nonzero idempotent EndR(N) and a homomorphism h : M (N) such that hf = , where : N N is the natural embedding. A module M is called almost injective if it is almost N-injective for every right R-module N. Dually, we define the concept of almost projective modules. A module M is called almost N-projective, if for any natural homomorphism g : N N/K and any homomorphism f : M N/K, either there exists a homomorphism h : M N such that f = gh or there exists a non-zero direct summand N of N and a homomorphism h : N M such that g = f h, where : N N is the natural embedding. A module M is called almost projective if it is almost N-projective for every right R-module N.
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The concepts of almost injective module and almost projective module were studied in the works [1]-[7] by Harada and his colleagues. Note that, in [7] an almost projective right R-module is defined as a module which is almost Nprojective to every finitely generated right R-module N. In recent years, almost injective modules were considered in [8]-[12]. The problem of the description of the rings over which all modules are almost injective was studied in [10]. In some special cases, this problem was solved in [10]. In particular, in the case of semiperfect rings. In this article, we study the structure of the rings over which every module is almost injective, in general. We also give the characterization of
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2010 Mathematics Subject Classification. 16D40, 16S50, 16S90. Key words and phrases. almost projective, almost injective modules, semiartinian rings, Vrings.
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1
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2
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ABYZOV A. N.
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the module M such that every simple module is almost projective (respectively, almost injective) in the category (M).
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Let M, N be right R-modules. We denote by (M) the full subcategory of Mod-R whose objects are all R-modules subgenerated by M. If N (M) then the injective hull of the module N in (M) will be denoted by EM (N). The Jacobson radical of the module M is denoted by J(M).
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The Loewy series of a module M is the ascending chain of submodules
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0 = Soc0(M ) Soc1(M ) = Soc(M ) . . . Soc(M ) Soc+1(M ) . . .,
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where Soc+1(M)/ Soc(M) = Soc(M/ Soc(M)) for all ordinal numbers and Soc(M) = Soc(M) for a limit ordinal number . Denote by L(M) the sub-
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<
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module Soc(M), where is the smallest ordinal such that Soc(M) = Soc+1(M). The module M is semiartinian if and only if M = L(M). In this case is called the Loewy length of module M and is denoted by Loewy(M). The ring R is called right semiartinian if the module RR is semiartinian.
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The present paper uses standard concepts and notations of ring theory (see, for example [13]-[15] ).
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1. Almost projective modules
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A module M is called an I0-module if every its nonsmall submodule contains nonzero direct summand of the module M.
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Theorem 1.1. For a module M, the following assertions are equivalent:
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1) Every simple module in the category (M) is almost projective. 2) Every module in the category (M) is either a semisimple module or con-
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tains a nonzero M-injective submodule. 3) Every module in the category (M) is an I0-module.
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Proof. 1)2) Let xR (M) be a non-semisimple cyclic module. Then the module xR contains an essential maximal submodule N. Let f : EM (xR) EM (xR)/N be the natural homomorphism and : xR/N EM (xR)/N be the embedding. Assume that there exists a homomorphism g : xR/N EM (xR) such that f g = . Since g(xR/N ) f -1(xR/N ) = xR and N is an essential submodule of xR, then g(xR/N) N. Consequently f g = 0, which is impossible. Since the module xR/N is almost projective, for some nonzero direct summand N of EM (xR) and homomorphism h : N xR/N we get h = f , where : N EM (xR) is the embedding. Consequently f (N ) xR/N, i.e. N f -1(xR/N ) = xR.
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2)3) The implication follows from [16, Theorem 3.4].
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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3
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3)1) Let S be a simple right R-module, f : A B be an epimorphism right R-modules and g : S B be a homomorphism. Without loss of generality, assume that g = 0. If Ker(f ) is not an essential submodule of f -1(g(S)), then there exists a simple submodule S of f -1(g(S)) such that f (S) = g(S). In this case, obviously, there is a homomorphism h : S A such that f h = g. Assume Ker(f ) is an essential submodule of f -1(g(S)). Then f -1(g(S)) is a non-semisimple module and by [16, Theorem 3.4], f -1(g(S)) contains a nonzero injective submodule A. There exists a homomorphism g : g(S) S such that gg(s) = s for all s g(S). Then g(gf|A) = f , where : A A is the embedding and f|A : A g(S) is the restriction of the homomorphism f to A.
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Corollary 1.1. Every right R-module is an I0-module if and only if every simple right R-module is almost projective.
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A right R-module M is called a V -module (or cosemisimple) if every proper submodule of M is an intersection of maximal submodules of M. A ring R is called a right V -ring if RR is a V -module. It is known that a right R-module M is a V -module if and only if every simple right R-module is M-injective. A ring R is called a right SV -ring if R is a right semiartinian right V -ring.
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Theorem 1.2. For a regular ring R, the following assertions are equivalent:
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1) Every right R-module is an I0-module. 2) R is a right SV -ring. 3) Every right R-module is almost projective. 4) Every simple right R-module is almost projective.
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Proof. The equivalence 1)2) follows from [16, theorem 3.7]. The implication 3)4) is obvious. The implication 4)1) follows from Theorem 1.1.
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2)3) Let S be a simple right R-module. We claim that the module S is almost projective. Let f : A B be an epimorphism right R-modules and g : S B be a homomorphism. Without loss of generality, assume that Ker(f ) = 0. Then Ker(f ) contains a simple injective submodule S and for the homomorphism h = 0 Hom(S, S) we get f = gh, where : S A is the natural embedding.
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A ring R is called a I-finite (or orthogonally finite) if it does not contain an infinite set of orthogonal nonzero idempotents.
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Theorem 1.3. For a I-finite ring R, the following assertions are equivalent:
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1) Every right R-module is almost projective. 2) Every simple right R-module is almost projective. 3) R is an artinian serial ring and J2(R) = 0.
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4
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ABYZOV A. N.
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Proof. The implicatio 1)2) is obvious.
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2)3) By Theorem 1.1 and [14, 13.58], R is a semiperfect ring. Then by [16, Theorem 3.2], R is an artinian serial ring and J2(R) = 0.
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3)1) Let M be a right R-module. We claim that the module M is almost
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projective. Let f : A B be an epimorphism of right R-modules and g : M B be a homomorphism. If f -1(g(M)) is a semisimple module, then it is obvious that there is a homomorphism h such that g = f h. Assume f -1(g(M)) is a non-semisimple module. Then the module f -1(g(M)) contains an injective and
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projective local submodule L of length two. Since L is a projective module, then there is a homomorphism h : L g-1(f (L)) such that f = g|g-1(f(L))h, where : L A is the natural embedding.
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2. Almost V -modules
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A right R-module M is called an almost V -module if every simple right Rmodule is almost N-injective for every module N (M). A ring R is called a right almost V -ring if every simple right R-module is almost injective. Right almost V -rings have been studied in [11].
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Lemma 2.1. For a module M, the following assertions are equivalent:
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1) M is not a V -module. 2) There exists a submodule N of the module M such that the factor mod-
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ule M/N is an uniform, Soc(M/N) is a simple module and M/N = S oc(M/N ).
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Proof. The implicatio 2)1) is obvious. 1)2) Since M is not a V -module, there is a submodule M0 such that J(M/M0) =
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0. Without loss of generality, assume that J(M/M0) contains a simple submodule S. Let S be a complement of submodule S in M/M0. Then (M/M0)/S is an uniform module, Soc((M/M0)/S) is a simple module and (M/M0)/S = Soc((M/M0)/S).
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Proposition 2.1. Let M be an almost V -module. Then:
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1) The Jacobson radical J(N) of every module N (M) is semisimple. 2) The factor module N/J(N) of every module N (M) is a V -module. 3) The injective hull EM (S) of every simple module S (M) is either a
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simple module or a local M-projective module of length two.
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Proof. 1) Assume that in the category (M) there exists a module whose Jacobson radical is not semisimple. Then there exists a module N (M) and a
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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5
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non-zero element x J(N) such that the module xR contains an essential max-
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imal submodule A. Let B be a complement of submodule A in N. Consider the
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homomorphism f : xR B xR/A is defined by f (xr + b) = xr + A, where
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r R, b B. Assume that there exists a homomorphism g : N xR/A such
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that g = f, where : xR B E is the natural embedding. Since x J(N),
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g(x) = 0. On the other hand, f (x) = 0. This is a contradiction. If there is a
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nonzero idempotent EndR(N) and a homomorphism h : xR/A (N) such that = hf, then hf ((N) (A B)) = 0 and ((N) (A B)) = 0 for a
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nonzero submodule (N) (A B), that is impossible. Thus a Jacobson radical
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J(N) of every module N (M) is semisimple.
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2) Let N (M) be a module and S (M) be a simple module, N0 be a submodule of N = N/J(N ) and f : N0 S be a homomorphism. We show that there exists a homomorphism g such that f = g, where : N0 N is the natural embedding. Without loss of generality, assume that N0 is essential in N and f = 0. Assume Ker(f ) is an essential submodule of N0. If there exists a nonzero idempotent EndR(N ) and a homomorphism h : S (N ) such that = hf, then hf ((N ) Ker(f )) = 0 and ((N ) Ker(f )) = 0 for a nonzero submodule (N)Ker(f ), that is impossible. Thus there exists a homomorphism
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g such that f = g. Assume Ker(f ) is not an essential submodule of N0. Then there exists a simple module S such that N0 = Ker(f ) S. Assume that there exists a non-zero idempotent EndR(N ) and a homomorphism h : S (N ) such that = hf. Since Ker(f ) S is essential in N , Ker(f ) (1 - )N and (1 - )N (S) = (1 - )N S, then (S) is essential in (N ). Since J(N ) = 0, we get (S) = (N ), and consequently N = (1 - )N S. Then there exists a g : (1-)N S S homomorphism is defined by g(n+s) = f (s), where n (1 - )N , s S such that f = g. Hence N is a V -module.
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3) Let S (M) be a simple module and EM (S) = S. By 2), J(EM (S)) = S. Let A1, A2 be maximal submodules of EM (S). From the proof of [17, 13.1(a)], we see that EndR(A1), EndR(A2) are local rings. Assume that A = Ai Aj is a CSmodule, where i, j {1, 2}. Let B is a closed submodule of A and A = B. Then B is complement of some simple submodule S in A. Consider the homomorphism f : S B S is defined by the formula f (s + b) = s, where s S, b B. Since S J(A) and S is an almost A-injective module, there is a non-zero idempotent
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EndR(A) and a homomorphism g HomR(S, (A)) such that gf = , where : S B A is the natural embedding. It's clear that B (1 - )A and
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S (1 -)A = 0. Consequently B = (1 -)A. Thus A is a CS-module. From [17,
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7.3(ii)] and the fact that every monomorphism : Ai Aj is an isomorphism we deduce that Ai is an Aj-injective module. If A1 = A2 then by [15, 16.2], A1 is an
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6
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ABYZOV A. N.
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A1 + A2-injective, which is impossible. Thus the module EM (S) has an unique maximal submodule, and consequently EM (S) is a local module of length two. We claim that EM (S) is projective in the category (M). Let N be a submodule of EM (S)M such that N +M = EM (S)M and : EM (S)M EM (S) be the natural projection. Assume that J(N) N M. Since N/J(N) is a V -module, (N) = EM (S) is a V -module, which is impossible. Thus there exists a simple submodule S of J(N ) such that SM = 0. Let A be a complement of submodule S in N such that M N A. Consider the homomorphism f : S A S is defined by f (s + a) = s, where s S, a A. Since S J(N ) and S is an almost N -injective module, there is a non-zero idempotent EndR(N ) and a homomorphism g HomR(S, (N )) such that gf = , where : S A N is the natural embedding. Since A (1 - )(N ), S J(N ) and A S is an essential submodule of N, we deduce that (S) is essential in (N ) and (S) = (N ). Since
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(N ) A = (N ) N M = (N ) M = 0
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and lg(EM (S)) = lg((N )) = 2, we have ((N )) = EM (S). Then (N ) M = EM (S) M. By [15, 41.14], the module EM (S) is projective in the category (M ).
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Theorem 2.1. For a module M, the following assertions are equivalent:
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1) M is an almost V -module. 2) Every module in the category (M) is either a V -module or contains a
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nonzero direct summand which is a projective object in the category (M). 3) There exist an independent set of local submodules {Ai}iI of the module
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M such that: a) Ai is both an M-injective and an M-projective module of length two for all i I; b) J(M ) = iI J(Ai); c) M/J(M) is a V -module.
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Proof. 1)2) Let N be a module in the category (M) which is not a V -module. Then by Lemma 2.1 and Proposition 2.1, there is a submodule N of N such that the factor module N/N is nonzero and projective in the category (M). Consequently the natural epimorphism f : N N/N splits and the module N
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contains a nonzero direct summand which is a projective in the category (M).
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2)1) Let M be a right R-module and S be a simple right R-module. We
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claim that S is an almost M-injective module. Let M0 be a submodule of M and f : M0 S be a homomorphism. Without loss of generality, assume that f = 0,
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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7
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M0 is an essential submodule of M and EM (S) = S. There is a homomorphism g : M EM (S) such that g = f, where : M0 M and : S EM (S) the natural embeddings. Assume that S = g(M). Then by the condition 2), g(M) is a projective module. Consequently M = Ker(g) M. Since M0 is an essential submodule of M, then M0 M is a simple module and f|M0M : M0 M S is an isomorphism. Then M0 = (M0 M) Ker(f ). Let : Ker(g) M M be the natural projection. Then = f|-M10Mf.
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1)3) By Zorn's Lemma, there is a maximal independent set of submodules {Ai}iI of the module M such that Ai is a local module of length two for all i I. According to Proposition 2.1, M/J(M) is a V -module and Ai is both an M-injective and an M-projective module for all i I. Assume that J(M) = iIJ(Ai). Then by the condition 1), there is a simple submodule S of M such that S J(M) and S iI J(Ai) = 0. Let S be a complement of submodule S in M such that it contains iI J(Ai). Then M/S is not a simple module, which is an essential extension of the simple module (S + S)/S. By Proposition 2.1, M/S is an M-projective module of length two. Consequently, there is a local submodule of length two L of M such that M = L S. This contradicts with the choice of the set {Ai}iI . Thus J(M ) = iI J(Ai).
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3)2) Let S (M) be a simple module and EM (S) = S. By [15, 16.3], there exists an epimorphism f : iIMi EM (S), where Mi = M for all i I. Since EM (S) is not a V -module, by [15, 23.4], f i(J(M)) = 0 for some i I, where i : Mi iIMi is a natural embedding. Then, by the conditions a) and b) of 3), EM (S) = Ai for some i I. Thus every essential extension of a simple module in the category (M) is either a simple or a local M-projective module
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of length two. Then the implication follows directly from Lemma 2.1.
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Corollary 2.1. For a ring R, the following assertions are equivalent:
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1) R is a right almost V -ring. 2) Each right R-module is either a V -module or contains a nonzero direct
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summand which is a projective module. 3) There exist a set of orthogonal idempotents {ei}iI of the ring R such that:
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a) eiR is a local injective right R-module of length two for every i I; b) J(P ) = iI J(eiR); c) R/J(R) is a right V -ring.
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Theorem 2.2. For a right noetherian ring R, the following assertions are equiv-
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alent:
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1) Every right R-module is a direct sum of an injective module and a V -
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module.
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8
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ABYZOV A. N.
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2) Every right R-moduleis a direct sum of a projective module and a V module.
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3) R is a right almost V -ring.
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Proof. 3)1), 2) By Zorn's Lemma, there is a maximal independent set of local submodules of length two {Li}iI of the module M. Since R is a right noetherian ring, by [13, 6.5.1], there exists a submodule N of M such that M = iILi N. By Proposition 2.1 3), iI Li is both injective and projective. We claim that N is a V -module. Assume that N is not a V -module. Then by the Proposition 2.1 3) and Lemma 2.1, there exists a factor module N/N0 of N which is a local projective module of length two. Consequently, the module N/N0 is isomorphic to a submodule of N, which contradicts the choice of the set {Li}iI. Thus N is a V -module.
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2)3) Let S be a right simple module. Assume that E(S) = S. By the condition 2), E(S) is a projective module and by [13, 7.2.8], EndR(E(S)) is a local ring. Then by [13, 11.4.1], E(S) is a local module. If J(E(S)) is not a simple module, then by the condition 2), the module E(S)/S is projective, and consequently S is a direct summand of E(S), which is impossible. Thus the injective hull of a every simple right R-module is either a simple or a projective module of length two. Consequently R is a right almost V -ring by [11, Theorem
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3.1]. 1)3) Since RR is a noetherian module then by the condition 1), RR = M N,
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where M is a finite direct sum of uniform injective modules and N is a V -module. By [13, 7.2.8, 11.4.1], M = L1 . . . Ln, where Li is a local module for every 1 i n. Assume that J(Li0) is nonzero and is not a simple module for some i0. Then there is a non-zero element r J(Li0) such that rR = J(Li0). Let T be maximal submodule of rR. By the condition 1), the injective hull of every simple right R-module is either a simple module or a module of length two. Then the local module Li0/T is not an injective module and it is not a V -module, which contradicts to condition 1). From these considerations, it follows that there exists a family of orthogonal idempotents e1, . . . en of ring R satisfying the condition a) and b) of Corollary 2.2, and RR/J(R) is the direct sum of a semisimple module and a V -module. By [15, 23.4], R/J(R) is a right V -ring. Then, by Corollary 2, R is an almost right V -ring.
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Theorem 2.3. For a regular ring R, the following assertions are equivalent:
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1) R is a right V -ring. 2) Every right R-module is a direct sum of an injective module and a V -
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module.
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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9
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3) Every right R-module is a direct sum of a projective module and a V -
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module.
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4) R is a right almost V -ring.
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Proof. The implications 1)2), 1)3), 1)4) are obvious.
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2)1) Assume that the ring R is not a right V -ring. Then E(S) = S for some simple right R-module S. By the condition 2) we have i=1Li = M N, where Li = E(S) for every i, M is an injective module and N is a V -module. Since J( i=1Li) is essential in i=1Li, then J( i=1Li) N = J(N ) is essential in N , and consequently N = 0. Let I = {r R | E(S)r = 0}. We can conside the module i=1Li as a right module over the ring R/I. Assume that R/I is not a semisimple artinian ring. Then the ring R/I contains a countable set of non-zero orthogonal idempotents {ei} i=1. For every i N, there is an element li Li such that liei = 0. Since the right R/I-module i=1Li is injective, there exists a homomorphism f : R/IR/I i=1Li, such that f (ei) = liei for all i. Since f (R/IR/I) ni=1Li for some n N, we obtain a contradiction with the fact that liei = 0 for all i N. Thus R/I is a semisimple artinian ring. Consequently E(S) = S. This contradiction shows that R is a right V -ring.
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3)1) Assume that the ring R is not a right V -ring. Then by Lemma 2.1, there
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exists a right ideal I of R such that the right R-module R/I is an uniform, is not a
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simple module and Soc(R/IR) is a simple module. Then, by the condition 3), the module R/I is projective, and consequently R/IR is isomorphic to a submodule of RR, which is impossible. This contradiction shows that R is a right V -ring.
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The implication 4)1) follows directly from Corollary 2.1.
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3. Rings Over Which Every Module Is Almost Injective
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Let M be a right R-module. Denote by SI(M) the sum of all simple injective submodules of the module M. Clearly, SI(RR) is ideal of ring R.
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Lemma 3.1. Let R be a ring with the following properties: a) in the ring R there exists a finite set of orthogonal idempotents {ei}iI such that eiR is local injective right R-module of length two, for each i I and J(R) = iI J(eiR); b) R/J(R) is a right SV -ring and Loewy(RR) 2; c) R/SI(RR) is a right artinian ring.
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Then we have the following statement: 1) the injective hull of every simple right R-module is either a simple module or a local projective module of length two;
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10
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ABYZOV A. N.
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2) every right R-module is a direct sum of a injective module and a V -module;
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3) every right R-module is a direct sum of a projective module and a V -
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module; 4) if S a simple submodule of the right R-module N, S J(N) and SN = 0
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for some submodule N of N, then there are submodules L, N of N such that L is a local module of length two, S L, N N and N = N L.
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Proof. 1) Let S be a simple right R-module and E(S) = S. Since R/J(R) is a right V -ring and J(R) is a semisimple right R-module, then E(S)S = 0 for some simple submodule of S of right R-module J(R)R. From condition a) it follows that S is essential in some injective local submodule of the module iIeiR. Therefore, E(S) = ei0R for some i0 I. Thus injective hull of every simple right R- module is either a simple module or a local projective module of length two.
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2), 3) Let M be a right R-module. By Lemma of Zorn there is a maximal independent set of submodules of {Li}iI of a module M such that Li is a local injective module of length two, for each i I. Clearly, E(iILi)SI(R) = 0. Then from the condition c) it follows that E(iI Li) = iI Li. Therefore M = iILi N for some submodule N of a module M . It is clear that module iILi is injective and projective. If N is V -module, then from Lemma 2.1 and condition 1) follows that for some submodule N0 of the module N factor module N/N0 is a local projective module of length two. Therefore N = N0 L where L is a injective local module of length two, which impossible. Thus N is a V -module.
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4) From conditions 1) and 2), it follows that S L where L is a local injective submodule of a module N of length two. Let L is a complement of L in N which contains the submodule N . Then (S + L)/L is a essential submodule of N/L and N/L = (S + L)/L. From condition 1), it follows that N/L is a local module of length two. Therefore, the natural homomorphism f : N N/L induces an isomorphism f|L : L N/L. Then N = L L.
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Lemma 3.2. Let M be a right R-module and N be a injective submodule of M. If N is submodule of M and N N = 0, then N N and M = N N for some submodule N of M
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Proof. Let M is a complement of N in M which contains the submodule N . Then E(M) = E(N ) N and M = (E(N ) M) N.
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Theorem 3.1. For a ring R the following conditions are equivalent:
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1) Every right R-module is almost injective. 2) R is a right semiartinian ring, Loewy(RR) 2 and every right R-module
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is a direct sum of an injective module and a V -module.
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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11
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3) R is a right semiartinian ring, Loewy(RR) 2 and every right R-module
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is a direct sum of a projective module and a V -module.
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4) The ring R satisfies the following conditions:
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a) in the ring R there exists a finite set of orthogonal idempotents {ei}iI such that eiR is a local injective right R-module of length two, for each
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i I and J(R) = iI J(eiR); b) R/J(R) is a right SV -ring and Loewy(RR) 2;
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c) R/SI(RR) is a right artinian ring. 5) The ring R is isomorphic to the ring of formal matrix
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T T MS 0S
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, where
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a) S is a right SV -ring and Loewy(S) 2;
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b) for some ideal I of a ring S the equality MI = 0 holds and the ring
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T T MS/I 0 S/I
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is an artinian serial, with the square of the Jacobson
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radical equal to zero.
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Proof. the Implication 4)2) and 4)3) follow from Lemma 2.
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1)4) From corollary 2.1 it follows that R/J(R) is a right V -ring. According to
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[10, proposition 2.6] Loewy(RR) 2. Then RR/Soc(RR) is a semisimple module of finite length, and from corollary 2.1 follows that the ring R contains a finite
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set of orthogonal idempotents {ei}iI satisfying the condition a) of 4). Therefore, RR = iIeiR A, where A is a semiartinian right R-module and Loewy(A) 2. As AJ(R) = 0, then, by corollary 2.1, A is a V -module. Suppose that Soc(A)
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contains an infinite family of primitive orthogonal idempotents {fi}iI such that fiR = E(fiR) for each i I. Let B is a complement of iIfiR in RR, which contains the J(R). Consider the homomorphism f : iIfiR B iIE(fiR), defined by f (r + b) = r, where r iIfiR, b B. Assume that : iIfiR B RR is a natural embedding. If there exists a homomorphism g : RR iIE(fiR) such that f = g then f (iIfiR) g(RR) iI E(fiR), where I I.Therefore | I |< , which is impossible. Since the module iIE(fiR) is a almost RR-injective, then there exists non-zero idempotent EndR(RR) and a homomorphism h : iIE(fiR) (RR) such that = hf. Since iIfiRB is essential in RR, then = 0. Therefore, h = 0. Then h(E(fi0R)) = 0 for some i0 I. Since (J(R)) = hf (J(R)) = 0, then J((RR)) = 0. From proposition 2.1 it follows that E(fi0R) is a local projective module of length two. Since J((RR)) = 0, then Ker(h|E(fi0,R)) and Im(h|E(fi0,R)) is a simple modules. Then Im(h|E(fi0,R)) is a direct summand of the module RR. Therefore, Ker(h|E(fi0R)) is a direct summand of the module E(fi0R), which is impossible. Thus, Soc(A) = SI(RR)
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12
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ABYZOV A. N.
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B where B is a module of finite length. Since A/Soc(A), Soc(A)/SI(RR) is a modules of finite length, then A/SI(RR) is a module of finite length. Therefore, R/SI(RR) is right artinian ring.
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4)1) Suppose that the ring R satisfy the condition 4) and M, N are right
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R-modules. We claim that M is an almost N-injective module. Let N0 is a submodule of N, and : N0 N be the natural embedding and f : N0 M is a homomorphism. Without loss of generality, we can assume that N0 is an essential submodule of N. In this case Soc(N ) = Soc(N0).
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Consider the following three cases.
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Case f (J(N) Soc(N)) = 0, f (SI(N)) = 0. There exists a homomorphism
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g : N E(M), such that the equality holds f = g. If g(N)SI(RR) = 0, then exists a primitive idempotent e R such that eR is a simple injective module and g(N)e = 0. Then neR is a simple injective module and f (neR) = g(neR) = 0 for
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some n N, which contradicts the equality f (SI(N)) = 0. Thus g(N)SI(RR) = 0. Since R/SI(RR) is a right Artinian ring and by Corollary 2.1, R/SI(RR) is an almost right V -ring, then by [10, Corollary 3.2], RR/SI(RR) is an Artinian serial ring and J2(RR/SI(RR)) = 0. Then by [14, 13.67], g(N ) = N1 N2, where N1 is a semisimple module and N2 is a direct sum of local modules of length two. If N2 = 0 then there exists an epimorphism h : N2 L, where L is a local module of length two. Since L is a projective module, hg is a split epimorphism,
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where : N1 N2 N2 is the natural projection. Consequently, hg|L is an isomorphism for some local submodule L of the module N and f (Soc(L)) = g(Soc(L)) = 0, which contradicts the equality f (J(N) Soc(N)) = 0. Then
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g(N) Soc(E(M)) M. Hence, we can conside the homomorphism g as an
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element of the Abelian group HomR(N, M). Case f (J(N) Soc(N)) = 0. If f (N0) is not a V -module, then by Lemma 2.1,
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there exists an epimorphism h : f (N0) L, where L is an uniform but is not a simple module, whose socle is a simple module. By lemma 3.1, L is a projective and injective module. Since L is a projective module, N0 = f -1(Ker(h)) L, f (N0) = Ker(h)f (L), where L is a submodule of N0 and L = L. By Lemma 3.2, the following conditions are satisfied for some direct summands M, N of
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modules M and N, respectively:
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M = M f (L), Ker(h) M , N = N L, f -1(Ker(h)) N .
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Let 1 : M f (L) f (L), 2 : N L L be natural projections. There exists an isomorphism h : f (L) L, such that f h = 1f(L). Then we have the equality (h1)f = 2.
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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13
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If f (N0) is a V -module, then for some simple submodule S of J(N)Soc(N) we have the equality f (N0) = f (S) M, where M is a submodule of the module M and f (S) = 0. Let : f (S) M f (S) be the natural projection. We can consider the homomorphism f as an element of the Abelian group HomR(N0, f (N0)). Then N0 = Ker(f ) S. By Lemma 3.1, the following conditions are satisfied for some submodules N and L of the module N :
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N = N L, Ker(f ) N , lg(L) = 2, Soc(L) = S.
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By Corollary 2.1, R is a right almost V -ring. Then by [11, 2.9], there exists a decomposition M = M1 M2 of module M, such that M1 is a complement for f (S) in M and M M1. Easy to see that 2(f (S)) is a simple essential submodule of M2, where 2 : M1 M2 M2 is the natural projection. Let h : S 2f (S) be the isomorphism is defined by h(s) = 2f (s) for every s S. We can consider the homomorphism h-1 as an element of the Abelian group HomR(2f (S), L). If M2 is a simple module, then we have the equality (h-12)f = , wher : N L L is the natural projection. If M2 is not a simple module, then since M2 is an injective module, there is an isomorphism h : M2 L such that h|2f(S) = h-1. Then we have the equality (h2)f = .
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Case f (SI(N)) = 0. In this case, for some simple injective submodule S of the module N we have f (S) = 0. Since f (S) is an injective module, M = f (S) M0, where M0 is a submodule of M. Let : f (S) M0 f (S) be the natural projection. Then N0 = Ker(f ) S. By Lemma 3.2, there exists a direct summand N of N such that:
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N = N S, Ker(f ) N .
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Let : N S S be the natural projection. There is an isomorphism h : f (S) S, such that f h = 1f(S). Then we have the equality (h)f = .
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2)4) Suppose that the ring R satisfy the condition 2). According to the condition 2), we have that R/J(R)R = A B, where A is an injective module and B is a V -module. By [18, Theorem 3.2], A has finite Goldie dimension. Since A is a semiartinian module and J(A) = 0, it follows that A is a semisimple module. Therefore, by [15, 23.4], R/J(R) is a V -ring.
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By the condition 2), this implies RR = A B, where A is an injective module and B is a V -module. It is easy to see, according to the condition 2), the injective hull of every simple R-module has the length at most 2. Then by [18, Theorem 3.2], A is a finite direct sum of modules of length at most 2.
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Let M be a right injective R/SI(R)-module and {Li}iI be a maximal independent set of submodules of M with lg(Li) = 2 for all i. By the condition 2),
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14
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ABYZOV A. N.
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iILi is an injective R/SI(R)-module. Consequently, there exists a submodule N of M such that M = iI Li N. If N (Soc(R)/SI(R)) = 0, then N contains a simple submodule S, such that S is not injective as right R-module. Then the
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injective hull E(S) of the right R-module S has the length two and obviously
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E(S)SI(R) = 0. Consequently, S is not a injective right R/SI(R)-module and
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there exists a local injective submodule L of N of length two such that S L. This
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contradicts the choice of the set {Li}iI. Consequently, N (Soc(R)/SI(R)) = 0 and since R/ Soc(R) is a Artinian semisimple ring, we have N is a semisimple
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module. Thus, every injective right R/SI(R)-module is a direct sum of injective
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hulls of simple modules and since [13, 6.6.4], we have that R/SI(RR) is a right
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Artinian ring. 3)4) Suppose that the ring R satisfy the condition 3). If R = R/J(R) is not
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a right V -ring, then by Lemma 2.1, there is a right ideal T of the ring R such that the right R-module R/T is an uniform but is not a simple module, whose socle is a simple module. Consequently, by the condition 3) the module R/T is projective and isomorphic to a submodule of RR , which is impossible. Hence, R/J(R) is a V -ring.
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Let S be a simple right R-module and E(S) = S. By condition 3), E(S)
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is a projective module. By [13, 7.2.8, 11.4.1], E(S) is a local module. Since
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Loewy(R) 2, it follows that E(S)/S is a semisimple module. Consequently,
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J(E(S)) = S and lg(E(S)) = 2.
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By Zorn's Lemma there is a maximal independent set of submodules {Li}iI of RR such that Li is a local injective module of length two for all i I. Since Loewy(RR) 2, it follows that I is a finite set and | I |< lg(RR/Soc(RR)). Then RR = iILi eR, where e2 = e R. By Lemma 2.1 and the condition 3), eR is a V -module. Consequently, J(R) = iI Li.
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Now assume that Soc(eR) contains an infinite set of orthogonal primitive idempotents {fi} i=1 with E(fiR) = fiR for all i. There exists a subset I of I, such that Z(Li) = 0 for all i I and f R = iI\ILi eR is a nonsingular module, where f 2 = f R. There exists a homomorphism f : RR E( i=1fiR) such that f (r) = r for all r i=1fiR. Since E( i=1fiR) is a nonsingular module, it is generated by the module iI\ILi eR. From the condition 3), implies that E( i=1fiR) is a projective module. Consequently, E( i=1fiR) can be considered as a direct summand of iIMi, where Mi = f R for all i I. There exists a finite subset {i1, . . . , ik} of I such that the following inclusion holds f (RR) Mi1 . . . Mik . Let : Mi1 . . . Mik (iI\{i1,...,ik}Mi) M iI\{i1,...,ik} i be the natural projection. Since iIMi is nonsingular and f (RR) is an essential submodule of E( i=1fiR), then (E( i=1fiR)) = 0. Then E( i=1fiR) is
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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15
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a direct summand of Mi1 . . . Mik , and consequently, E( i=1fiR) is finitely generated. By [18, Theorem 3.2], E( i=1fiR) has finite Goldie dimension, which
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is impossible. Thus, Soc(eR) = SI(R) S, where S is a semisimple module of
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finite length. Consequently, R/SI(RR) is a right Artinian ring.
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4)5) Suppose that the ring R satisfy the condition 4). There is an idem-
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potent e R such that eR = iIeiR. It is clear that eRSI(R) = 0 and
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SI(R) (1 -e)R. By the condition 4), (1 -e)R is a semiartinian V -module, then (1 - e)Re = 0. Easy to see that (1 - e)R/J(R)(1 - e) = (1 - e)R(1 - e), where e = e + J(R). By [19, Theorem 2.9], (1 - e)R/J(R)(1 - e) = EndR/J(R)(1 - e)R/J(R)
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is a right SV -ring and Loewy((1-e)R/J(R)(1-e)) 2. Thus, the Peirce decom-
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position
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eRe eReeR(1 - e)(1-e)R(1-e)
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0
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(1 - e)R(1 - e)
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of the ring R satisfies the conditions
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a) and b) of 4). By Lemma 3.1, every right module over the ring R/SI(R)
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is a direct sum of an injective module and a V -module. It is clear that ev-
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ery V -module over a right Artinian ring is semisimple, then, by [14, 13.67],
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R/SI(R) =
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eRe eReeR(1 - e)(1-e)R(1-e)/SI(R) 0 (1 - e)R(1 - e)/SI(R)
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is an Artinian serial ring
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whose the square of the Jacobson radical is zero.
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5)4) Put
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R =
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T T MS 0S
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,I =
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00 0I
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,e =
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10 00
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,f =
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00 01
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.
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Since eRI = 0 and R/I is an Artinian serial ring whose the square of the
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Jacobson radical is zero, there exists a finite set of orthogonal idempotents {ei}iI and a semisimple submodule A of RR such that eRR = iI eiR A, and for every i eiR is a local right R-module of length two and eiR as right R/Imodule is injective. We claim that eiR is an injective R-module for every i. Suppose that E(eiR)I = 0. Then, there exists an elements r I, m E(eiR) such that mrR = Soc(eiR). Since eiRI = 0 and S is a regular ring, then Soc(eiR) = mrR = mrRrR = 0. This is a contradiction. Thus, eiR is an injective module for every i. Since S = R/eR is a right V -ring and f ReR = 0, we have f R is a V -module. Since
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RR = iI eiR A f R
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and Af R is a V -module, we have that J(R) = iI J(eiR) and Loewy(RR ) 2. Since R/J(R) = T /J(T ) <20> S, it follows that R/J(R) is a right SV -ring.
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There exists a right ideal I of R such that
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Soc(f RR ) = I (Soc(f RR ) I).
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16
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ABYZOV A. N.
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Since the right R-module I isomorphic to the submodule f RR /I and lg(f RR /I) < , we have that lg(I) < . Let N is a simple submodule of Soc(f RR ) I. We show that N is an injective module. Assume that E(N)e = 0. Then, there
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exists elements r R, n E(N ) such that nerR = N. Since eRI = 0 and S is a regular ring, we have N I = N , and consequently N = N I = nerRI = 0,
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which is impossible. Thus NeR = 0. Consequently, we can consider N as a module over the ring R/eR. Since R/eR = S is a right V -ring, it follows that E(N ) = N. Thus Soc(f RR ) I = Soc(I) SI(R). Since R/ Soc(R)R is a semisimple module and I/ Soc(I)R isomorphic to a submodule of R/ Soc(R)R, we have that I/ Soc(I)R is a module of finite length. Since R/IR , I/ Soc(I)R are modules of finite length, we have R/ Soc(I)R is a module of finite length. Consequently, R/SI(R) is a right Artinian ring.
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Theorem 3.2. For a ring R the following conditions are equivalent:
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1) Every R-module is almost injective. 2) The ring R is a direct product of the SV -ring whith Loewy(RR) 2, and
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an artinian serial ring, with the square of the Jacobson radical equal to zero.
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Proof. The implication 2)1) follows from the previous theorem.
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1)2) According to Theorem 3.1, the ring R isomorphic to the formal upper
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triangular matrix ring R =
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TM 0S
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, satisfying the conditions of Theorem 3.1.
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5). Since every left R-module is almost injective, from the analogue of Theorem
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7 on the left-hand side, it implies that J(R) contained in a finite direct sum of
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left local injective R-modules of length two. Since M =
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0M 00
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J(R),
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it follows that M = J(
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n i=1
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Rei)
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=
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n i=1
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J
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(R)ei
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,
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where
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e1, . . . , en
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are
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or-
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thogonal primitive idempotents and Rei is a local injective module of length
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two for every 1 i n. For every 1 i n, the idempotent ei has
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the form
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fi mi 0 ei
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, where fi, ei are idempotents respectively rings T and
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S. Since J(R)
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fi mi 0 ei
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=
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J(T )fi M ei
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0
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0
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is a simple submodule of the
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left R-module M, it follows that Mei = 0, and consequently ei = 0. Since
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ei + J(R) is a primitive idempotent of the ring R/J(R), we have fi = 0. Thus,
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ei =
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0 mi 0 ei
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, where ei is a primitive idempotent of the ring S and miei = mi.
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ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
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17
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Since M =
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0M 00
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=
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n i=1
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J
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(R)ei
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,
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then
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M
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=
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ni=1M ei
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is
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a
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decomposi-
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tion of the semisimple left T -module into a direct sum of simple submodules
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and M(1 - S such that
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eSni==1 eie)iS=,
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0. If there exists where 1 i n,
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a primitive idempotent e then Me = 0. Then the
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of the ring right ideals
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(
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n i=1
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ei)S
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and
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(1
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-
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n i=1
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ei)S
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of
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the
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ring
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S
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do
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not
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contain
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isomorphic
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simple
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right R-submodules. Consequently, e =
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n i=1
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ei
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is
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a
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central
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idempotent
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of
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the
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ring S and the ring R is isomorphic to the direct product of the SV -ring (1 - e)S
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and the Artinian serial ring
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TM 0 eS
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whose the square of the Jacobson radical
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is zero.
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The following theorem follows from the previous theorem and [20, theorem 1.7].
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Theorem 3.3. For commutative rings R the following conditions are equivalent:
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1) Every R-module is almost injective; 2) Every R-module is an extension of the semisimple module by an injective
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one.
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References
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[1] Y. Baba, Note on almost M-injectives, Osaka J. Math. 26(1989) 687698 [2] M. Harada, T. Mabuchi, On almost M-projectives, Osaka J. Math. l 26(1989) 837848 [3] Y. Baba, M. Harada, On almost M-projectives and almost M-injectives, Tsukuba J. Math.
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14(1990) 5369 [4] M. Harada, On almost relative injectives on Artinian modules, Osaka J. Math. 27(1990)
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963971 [5] M. Harada, Direct sums of almost relative injective modules, Osaka J. Math. 28(1991)
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751758 [6] M. Harada, Note on almost relative projectives and almost relative injectives, Osaka J.
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Math. 29(1992) 435446 [7] M. Harada, Almost projective modules, J. Algebra 159(1993) 150157 [8] A. Alahmadi, S. K. Jain, A note on almost injective modules, Math. J. Okayam, 51(2009)
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101-109 [9] A. Alahmadi, S. K. Jain, S. Singh, Characterizations of Almost Injective Modules, Con-
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temp. Math. 634(2015) 11-17 [10] M. Arabi-Kakavand, S. Asgari, Y. Tolooe, Rings Over Which Every Module Is Almost
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Injective, Communications in Algebra 44(7)(2016) 2908-2918 [11] M. Arabi-Kakavand, S. Asgari, H. Khabazian, Rings for which every simple module is
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almost injective, Bull. Iranian Math. Soc. 42(1)(2016) 113-127 [12] S. Singh, Almost relative injective modules, Osaka J. Math. 53( 2016) 425438 [13] Kasch, F. Modules and Rings, Academic Press 1982.
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18
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ABYZOV A. N.
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[14] A.A. Tuganbaev, Ring Theory. Arithmetical Modules and Rings, MCCME, Moscow, 2009, 472 .
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[15] R. Wisbauer, Foundations of Module and Ring Theory Philadelphia: Gordon and Breach 1991
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[16] A. N. Abyzov, Weakly regular modules over normal rings, Sibirsk. Mat. Zh., 49:4 (2008), 721738
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[17] N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer, Extending Modules, Longman, Harlow Pitman Research Notes in Mathematics 313 1994
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[18] H. Q. Dinh, D. V. Huynh, Some results on self-injective rings and - CS rings, Commun. Algebra 31(12)(2003) 60636077
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[19] G. Baccella, Semi-Artinian V-rings and semi-Artinian von Neumann regular rings, J. Algebra 173(1995) 587612
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[20] A. N. Abyzov, Regular semiartinian rings, Russian Mathematics (Izvestiya VUZ. Matematika), 2012, 56:1, 18
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[21] P. A. Krylov, A. A. Tuganbaev, Modules over formal matrix rings, Fundament. i prikl. matem., 15:8 (2009), 145211
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Department of Algebra and Mathematical Logic, Kazan (Volga Region) Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia
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E-mail address: aabyzov@ksu.ru
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