Article
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
Department of Mathematics, University of Guilan, P. O. Box 1914, Rasht, Iran
e-mail : ebrahimi@guilan.ac.ir, mehdikhoramdel@gmail.com and saboura_dolati@yahoo.com
Received: June 20, 2019; Revised: August 5, 2020; Accepted: August 18, 2020
Abstract
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
1. Introduction
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 [11] 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
and
The following are used in the sequel.
Proposition 1.1.
(i)([13, Proposition 1.5]) Let
M be an amply supplemented module. Then every submodule ofM has an s-closure.(ii)([4, 3.7(6)]) Let
M be anR -module andK≤ L≤ M . If, then and the converse is true if . (iii)([13, 31.4(2)]) Let
M be an amply supplemented module andB ≤ C submodules ofM such thatC/B is co-closed inM/B andB is co-closed inM . ThenC is co-closed inM .(iv)([2, Lemma 1.1]) If
M=⊕i∈ IMi andN is a fully invariant submodule of M, thenN=⊕i∈ I(N∩ Mi) .
Theorem 1.2. ([8, 15])
The following are equivalent for an
(1)
M is strongly lifting; (2) For each
N ≤ M , there existse∈ Sl(S) such thateM⊆ N and(1-e)M∩ N≪ (1-e)M .(3)
M is lifting and each direct summand of M is fully invariant. (4)
M is lifting and S is Abelian. (5)
M is amply supplemented and each coclosed submodule is a fully invariant direct summand in M .
Proposition 1.3. ([8, 15])
Let
Theorem 1.4. ([8, 15])
If
Lemma 1.5. ([8, 15])
Let
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.
Example 2.1.
(1) Let
, and Then M1 andM2 are strongly liftingR -modules, howeverR=M1 ⊕ M2 is not strongly lifting, by Theorem 1.2.(2) Let
R be a uniserial ring andI a nonzero proper right ideal ofR . ThenR andI are strongly liftingR -modules. IfR ⊕ I is strongly lifting, thenR ⊕ I has SSSP. So it is a C3-module. This implies thatI≤⊕ R , by [7, Corollary 3.2], which is a contradiction.
Proposition 2.2.
(i) Let
M=⊕i∈ IMi . IfM is a strongly lifting module, thenHom(Mi,Mj)=0 for eachi≠ j ofI .
(ii) A free
R -moduleF is strongly lifting if and only ifRR is strongly lifting and rank(F )=1.
(ii) Assume that
Theorem 2.3.
(i) Let
M be anR -module. The following are equivalent:
(1)
M is strongly lifting;(2) If
M=N+K (N,K ≤ M ), then there exists a fully invariant direct summandX sayM=X ⊕ Y such thatX ⊆ N ,Y ⊆ K andY ∩ N ≪ Y .
(ii) Let
M be a strongly lifting module. ThenM is π-projective.
(ii) is clear from (i).
Theorem 2.4.
Let
Now, let
Corollary 2.5.
Let
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.
Example 2.6.
Let
We now define a relative version of particular projective condition which is useful in our main theorems.
Definition 2.7.
Let
This is obviously true if and only if
Lemma 2.8.
(i) Let
M andN be two modules. IfM isN -strongly projective and, then M is-strongly projective and -strongly projective. (ii)
is N -strongly projective if and only ifMi isN -strongly projective for eachi ∈ I .
(iii)
M is-strongly projective if and only if M isNi -strongly projective for each1 ≤ i ≤ n .
(iv) Let
M be a finitely generated module. ThenM is⊕i ∈ INi -strongly projective if and only ifM isNi -strongly projective for eachi∈ I .
(ii) Let
(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.
Example 2.9.
It is clear that
Lemma 2.10.
Let
(1)
M2 isM1 -strongly projective;
(2) If
M=K+M1 for someK ≤ M , thenM2 ⊆ K .
In the following theorem, we present necessary and sufficient conditions under which direct sum of finite strongly lifting modules is strongly lifting.
Theorem 2.11.
Let
(i)
M1 andM2 are relatively strongly projective.
(ii)
M1 andM2 are strongly lifting.
Conversely, let $K$ be a coclosed submodule of $M$ such that $K+M_1=M$. By Lemma 2.10,
Corollary 2.12.
Let
Conversely, assume that
In [5], it is shown an
Theorem 2.13.
Let
Corollary 2.14.
Let
In the next theorem, it is considered when direct sum of arbitrary hollow modules is strongly lifting.
Theorem 2.15.
Let
(1)
M is strongly lifting;(2)
Mi is⊕j ∈ I, j ≠ iMj -strongly projective.
In the following, it is considered when direct sum of an arbitrary local modules is strongly lifting.
Corollary 2.16.
Let
Conversely, let
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.
Theorem 2.17
Let
Acknowledgements.
We would like to thank the referees for valuable comments.
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