Kyungpook Mathematical Journal 2020; 60(4): 673-682
Published online December 31, 2020
Copyright © Kyungpook Mathematical Journal.
Direct Sums of Strongly Lifting Modules
Shahabaddin Ebrahimi Atani, Mehdi Khoramdel* and Saboura Dolati Pishhesari
Received: June 20, 2019; Revised: August 5, 2020; Accepted: August 18, 2020
For the recently defined notion of strongly lifting modules, it has been shown that a direct sum is not, in general, strongly lifting. In this paper we investigate the question: When are the direct sums of strongly lifting modules, also strongly lifting? We introduce the notion of a relatively strongly projective module and use it to show if M=M1⊕ M2 is amply supplemented, then M is strongly lifting if and only if M1 and M2 are relatively strongly projective and strongly lifting. Also, we consider when an arbitrary direct sum of hollow (resp. local) modules is strongly lifting.
Keywords: lifting modules, strongly lifting modules, coclosed submodules, hollow modules, relatively strongly projective modules
Supplemented and lifting modules are worthy of study in module theory since they are the duals of complemented and extending modules. A number of results concerning lifting modules have appeared in the literature in recent years. Lifting modules were first introduced by Takeuchi  but under the name codirect modules. An
It is of natural interest to investigate whether or not an algebraic notion for modules is inherited by direct summands and direct sums. The purpose of this paper is to study the direct sum of strongly lifting modules. The direct sum of two strongly lifting modules need not be strongly lifting. We look at when direct sums of finitely many strongly lifting modules are strongly lifting. It is shown that every strongly lifting module is a direct sum of hollow modules and every strongly lifting module is π-projective and a direct sum of hollow modules. We introduce the notion of relatively strongly projective modules and use it to show if
Throughout, all rings (not necessarily commutative rings) have identity and all modules are unital right modules. For completeness, we now state some definitions and notations used in this paper. Let
The following are used in the sequel.
(i)([13, Proposition 1.5]) Let
Mbe an amply supplemented module. Then every submodule of Mhas an s-closure.
(ii)([4, 3.7(6)]) Let
Mbe an R-module and K≤ L≤ M. If , then and the converse is true if .
(iii)([13, 31.4(2)]) Let
Mbe an amply supplemented module and B ≤ Csubmodules of Msuch that C/Bis co-closed in M/Band Bis co-closed in M. Then Cis co-closed in M.
(iv)([2, Lemma 1.1]) If
M=⊕i∈ IMiand Nis a fully invariant submodule of M, then N=⊕i∈ I(N∩ Mi).
The following are equivalent for an
M is strongly lifting;
(2) For each
N ≤ M, there exists e∈ Sl(S)such that eM⊆ Nand (1-e)M∩ N≪ (1-e)M.
M is lifting and each direct summand of M is fully invariant.
M is lifting and S is Abelian.
M is amply supplemented and each coclosed submodule is a fully invariant direct summand in M.
2. Direct Sums of Strongly Lifting Modules
This section is devoted to investigate when direct sums of strongly lifting modules are strongly lifting. While it was shown in [8, 15] that every direct summand of a strongly lifting module is always strongly lifting, the following examples show that in general, the direct sum of strongly lifting modules is not a strongly lifting module.
, and Then M1and M2are strongly lifting R-modules, however R=M1 ⊕ M2is not strongly lifting, by Theorem 1.2.
Rbe a uniserial ring and Ia nonzero proper right ideal of R. Then Rand Iare strongly lifting R-modules. If R ⊕ Iis strongly lifting, then R ⊕ Ihas SSSP. So it is a C3-module. This implies that I≤⊕ R, by [7, Corollary 3.2], which is a contradiction.
M=⊕i∈ IMi. If Mis a strongly lifting module, then Hom(Mi,Mj)=0for each i≠ jof I.
(ii) A free
R-module Fis strongly lifting if and only if RRis strongly lifting and rank( F)=1.
(ii) Assume that
Mbe an R-module. The following are equivalent:
Mis strongly lifting;
M=N+K( N,K ≤ M), then there exists a fully invariant direct summand Xsay M=X ⊕ Ysuch that X ⊆ N, Y ⊆ Kand Y ∩ N ≪ Y.
Mbe a strongly lifting module. Then Mis π-projective.
(ii) is clear from (i).
By Corollary 2.5, we can see that every strongly lifting ring (considered as a module over itself) is semiperfect, the following example shows that the converse is not true, in general.
We now define a relative version of particular projective condition which is useful in our main theorems.
This is obviously true if and only if
Mand Nbe two modules. If Mis N-strongly projective and , then Mis -strongly projective and -strongly projective.
is N-strongly projective if and only if Miis N-strongly projective for each i ∈ I.
Mis -strongly projective if and only if Mis Ni-strongly projective for each 1 ≤ i ≤ n.
Mbe a finitely generated module. Then Mis ⊕i ∈ INi-strongly projective if and only if Mis Ni-strongly projective for each i∈ I.
(iii) The necessity is clear by (i). For the sufficiency, it is sufficient to show that if
(iv) Assume that
The result of Lemma 2.8(iii) does not extend to infinite direct sums, as the following example shows.
It is clear that
M2is M1-strongly projective;
M=K+M1for some K ≤ M, then M2 ⊆ K.
In the following theorem, we present necessary and sufficient conditions under which direct sum of finite strongly lifting modules is strongly lifting.
M1and M2are relatively strongly projective.
M1and M2are strongly lifting.
Conversely, let $K$ be a coclosed submodule of $M$ such that $K+M_1=M$. By Lemma 2.10,
Conversely, assume that
In , it is shown an
In the next theorem, it is considered when direct sum of arbitrary hollow modules is strongly lifting.
Mis strongly lifting;
Miis ⊕j ∈ I, j ≠ iMj-strongly projective.
In the following, it is considered when direct sum of an arbitrary local modules is strongly lifting.
In the next theorem, we use Theorem 2.15 to show that each factor module by a fully invariant submodule of a strongly lifting module is strongly lifting.
We would like to thank the referees for valuable comments.
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