arXiv:1701.00026v1 [math.RA] 30 Dec 2016

ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES
ABYZOV A. N.
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.
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.
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
2010 Mathematics Subject Classification. 16D40, 16S50, 16S90. Key words and phrases. almost projective, almost injective modules, semiartinian rings, Vrings.
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the module M such that every simple module is almost projective (respectively, almost injective) in the category (M).
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).
The Loewy series of a module M is the ascending chain of submodules
0 = Soc0(M )  Soc1(M ) = Soc(M )  . . .  Soc(M )  Soc+1(M )  . . .,
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-
<
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.
The present paper uses standard concepts and notations of ring theory (see, for example [13]-[15] ).

1. Almost projective modules
A module M is called an I0-module if every its nonsmall submodule contains nonzero direct summand of the module M.
Theorem 1.1. For a module M, the following assertions are equivalent:
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-
tains a nonzero M-injective submodule. 3) Every module in the category (M) is an I0-module.
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.
2)3) The implication follows from [16, Theorem 3.4].

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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.
Corollary 1.1. Every right R-module is an I0-module if and only if every simple right R-module is almost projective.
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.
Theorem 1.2. For a regular ring R, the following assertions are equivalent:
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.
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.
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.
A ring R is called a I-finite (or orthogonally finite) if it does not contain an infinite set of orthogonal nonzero idempotents.
Theorem 1.3. For a I-finite ring R, the following assertions are equivalent:
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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Proof. The implicatio 1)2) is obvious.
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.
3)1) Let M be a right R-module. We claim that the module M is almost
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
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.

2. Almost V -modules
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].
Lemma 2.1. For a module M, the following assertions are equivalent:
1) M is not a V -module. 2) There exists a submodule N of the module M such that the factor mod-
ule M/N is an uniform, Soc(M/N) is a simple module and M/N = S oc(M/N ).
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) =
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).
Proposition 2.1. Let M be an almost V -module. Then:
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
simple module or a local M-projective module of length two.
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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non-zero element x  J(N) such that the module xR contains an essential max-
imal submodule A. Let B be a complement of submodule A in N. Consider the
homomorphism f : xR  B  xR/A is defined by f (xr + b) = xr + A, where
r  R, b  B. Assume that there exists a homomorphism g : N  xR/A such
that g = f, where  : xR  B  E is the natural embedding. Since x  J(N),
g(x) = 0. On the other hand, f (x) = 0. This is a contradiction. If there is a
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
nonzero submodule (N)  (A  B), that is impossible. Thus a Jacobson radical
J(N) of every module N  (M) is semisimple.
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
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.
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
  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
S (1 -)A = 0. Consequently B = (1 -)A. Thus A is a CS-module. From [17,
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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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
(N )  A = (N )  N  M = (N )  M = 0
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 ).

Theorem 2.1. For a module M, the following assertions are equivalent:
1) M is an almost V -module. 2) Every module in the category (M) is either a V -module or contains a
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
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.

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
contains a nonzero direct summand which is a projective in the category (M).
2)1) Let M be a right R-module and S be a simple right R-module. We
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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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.
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).
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
of length two. Then the implication follows directly from Lemma 2.1.
Corollary 2.1. For a ring R, the following assertions are equivalent:
1) R is a right almost V -ring. 2) Each right R-module is either a V -module or contains a nonzero direct
summand which is a projective module. 3) There exist a set of orthogonal idempotents {ei}iI of the ring R such that:
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.
Theorem 2.2. For a right noetherian ring R, the following assertions are equiv-
alent:
1) Every right R-module is a direct sum of an injective module and a V -
module.

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2) Every right R-moduleis a direct sum of a projective module and a V module.
3) R is a right almost V -ring.

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.
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
3.1]. 1)3) Since RR is a noetherian module then by the condition 1), RR = M N,
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.

Theorem 2.3. For a regular ring R, the following assertions are equivalent:
1) R is a right V -ring. 2) Every right R-module is a direct sum of an 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 -
module.
4) R is a right almost V -ring.
Proof. The implications 1)2), 1)3), 1)4) are obvious.
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.
3)1) Assume that the ring R is not a right V -ring. Then by Lemma 2.1, there
exists a right ideal I of R such that the right R-module R/I is an uniform, is not a
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.
The implication 4)1) follows directly from Corollary 2.1.

3. Rings Over Which Every Module Is Almost Injective
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.
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.
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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2) every right R-module is a direct sum of a injective module and a V -module;
3) every right R-module is a direct sum of a projective module and a V -
module; 4) if S a simple submodule of the right R-module N, S  J(N) and SN = 0
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.

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.
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.
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.
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
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.

Theorem 3.1. For a ring R the following conditions are equivalent:
1) Every right R-module is almost injective. 2) R is a right semiartinian ring, Loewy(RR)  2 and every right R-module
is a direct sum of an injective module and a V -module.

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3) R is a right semiartinian ring, Loewy(RR)  2 and every right R-module

is a direct sum of a projective module and a V -module.

4) The ring R satisfies the following conditions:

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

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. 5) The ring R is isomorphic to the ring of formal matrix

T T MS 0S

, where

a) S is a right SV -ring and Loewy(S)  2;

b) for some ideal I of a ring S the equality MI = 0 holds and the ring

T T MS/I 0 S/I

is an artinian serial, with the square of the Jacobson

radical equal to zero.

Proof. the Implication 4)2) and 4)3) follow from Lemma 2.
1)4) From corollary 2.1 it follows that R/J(R) is a right V -ring. According to
[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
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)
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)

12

ABYZOV A. N.

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.
4)1) Suppose that the ring R satisfy the condition 4) and M, N are right
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).
Consider the following three cases.
Case f (J(N)  Soc(N)) = 0, f (SI(N)) = 0. There exists a homomorphism
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
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,
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
g(N)  Soc(E(M))  M. Hence, we can conside the homomorphism g as an
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,
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
modules M and N, respectively:

M = M   f (L), Ker(h)  M , N = N   L, f -1(Ker(h))  N .

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.

ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES

13

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 :
N = N   L, Ker(f )  N , lg(L) = 2, Soc(L) = S.

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 = .
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:
N = N   S, Ker(f )  N .
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 = .
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.
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.
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),

14

ABYZOV A. N.

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
injective hull E(S) of the right R-module S has the length two and obviously
E(S)SI(R) = 0. Consequently, S is not a injective right R/SI(R)-module and
there exists a local injective submodule L of N of length two such that S  L. This
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
module. Thus, every injective right R/SI(R)-module is a direct sum of injective
hulls of simple modules and since [13, 6.6.4], we have that R/SI(RR) is a right
Artinian ring. 3)4) Suppose that the ring R satisfy the condition 3). If R = R/J(R) is not
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.
Let S be a simple right R-module and E(S) = S. By condition 3), E(S)
is a projective module. By [13, 7.2.8, 11.4.1], E(S) is a local module. Since
Loewy(R)  2, it follows that E(S)/S is a semisimple module. Consequently,
J(E(S)) = S and lg(E(S)) = 2.
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.
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

ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES

15

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
is impossible. Thus, Soc(eR) = SI(R)  S, where S is a semisimple module of

finite length. Consequently, R/SI(RR) is a right Artinian ring.

4)5) Suppose that the ring R satisfy the condition 4). There is an idem-

potent e  R such that eR = iIeiR. It is clear that eRSI(R) = 0 and

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)
is a right SV -ring and Loewy((1-e)R/J(R)(1-e))  2. Thus, the Peirce decom-

position

eRe eReeR(1 - e)(1-e)R(1-e)

0

(1 - e)R(1 - e)

of the ring R satisfies the conditions

a) and b) of 4). By Lemma 3.1, every right module over the ring R/SI(R)

is a direct sum of an injective module and a V -module. It is clear that ev-

ery V -module over a right Artinian ring is semisimple, then, by [14, 13.67],

R/SI(R) =

eRe eReeR(1 - e)(1-e)R(1-e)/SI(R) 0 (1 - e)R(1 - e)/SI(R)

is an Artinian serial ring

whose the square of the Jacobson radical is zero.

5)4) Put

R =

T T MS 0S

,I =

00 0I

,e =

10 00

,f =

00 01

.

Since eRI = 0 and R/I is an Artinian serial ring whose the square of the
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

RR  = iI eiR  A  f R
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 ) × S, it follows that R/J(R) is a right SV -ring.
There exists a right ideal I of R such that

Soc(f RR ) = I  (Soc(f RR )  I).

16

ABYZOV A. N.

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
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,
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.

Theorem 3.2. For a ring R the following conditions are equivalent:
1) Every R-module is almost injective. 2) The ring R is a direct product of the SV -ring whith Loewy(RR)  2, and
an artinian serial ring, with the square of the Jacobson radical equal to zero.

Proof. The implication 2)1) follows from the previous theorem.

1)2) According to Theorem 3.1, the ring R isomorphic to the formal upper

triangular matrix ring R =

TM 0S

, satisfying the conditions of Theorem 3.1.

5). Since every left R-module is almost injective, from the analogue of Theorem

7 on the left-hand side, it implies that J(R) contained in a finite direct sum of

left local injective R-modules of length two. Since M =

0M 00

 J(R),

it follows that M = J(

n i=1

Rei)

=

n i=1

J

(R)ei

,

where

e1, . . . , en

are

or-

thogonal primitive idempotents and Rei is a local injective module of length

two for every 1  i  n. For every 1  i  n, the idempotent ei has

the form

fi mi 0 ei

, where fi, ei are idempotents respectively rings T and

S. Since J(R)

fi mi 0 ei

=

J(T )fi M ei

0

0

is a simple submodule of the

left R-module M, it follows that Mei = 0, and consequently ei = 0. Since

ei + J(R) is a primitive idempotent of the ring R/J(R), we have fi = 0. Thus,

ei =

0 mi 0 ei

, where ei is a primitive idempotent of the ring S and miei = mi.

ALMOST PROJECTIVE AND ALMOST INJECTIVE MODULES

17

Since M  =

0M 00

=

n i=1

J

(R)ei

,

then

M

=

ni=1M ei

is

a

decomposi-

tion of the semisimple left T -module into a direct sum of simple submodules

and M(1 - S such that

eSni==1 eie)iS=,

0. If there exists where 1  i  n,

a primitive idempotent e then Me = 0. Then the

of the ring right ideals

(

n i=1

ei)S

and

(1

-

n i=1

ei)S

of

the

ring

S

do

not

contain

isomorphic

simple

right R-submodules. Consequently, e =

n i=1

ei

is

a

central

idempotent

of

the

ring S and the ring R is isomorphic to the direct product of the SV -ring (1 - e)S

and the Artinian serial ring

TM 0 eS

whose the square of the Jacobson radical

is zero.

The following theorem follows from the previous theorem and [20, theorem 1.7].

Theorem 3.3. For commutative rings R the following conditions are equivalent:
1) Every R-module is almost injective; 2) Every R-module is an extension of the semisimple module by an injective
one.

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ABYZOV A. N.

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Department of Algebra and Mathematical Logic, Kazan (Volga Region) Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia
E-mail address: aabyzov@ksu.ru

