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Rings whose injective hulls are dual square free. (English) Zbl 1444.16003

Commun. Algebra 48, No. 3, 1011-1021 (2020); correction ibid. 48, No. 6, 2723 (2020).
Recall that a module \(M\) is called square-free if \(M\) contains no nonzero isomorphic submodules \(A\) and \(B\) with \(A\cap B=0\). Dually a module \(M\) is called dual square-free (DSF-module) if \(M\) has no proper submodules \(A\) and \(B\) such that \(M=A+B\) and \(M/A\cong M/B\).
Let \(R\) be a ring and \(E(R_R)\) be the injective hull of \(R_R\). In this paper, the authors study certain classes of rings \(R\) for which \(E(R_R)\) is a DSF-module. For example, it is shown that if \(R\) is right hereditary such that \(E(R_R)\) is a DSF-module, then \(R\) is right Noetherian, right distributive, right duo, and every subfactor of \(R_R\) is quasi-continuous. Also it is shown that if \(R\) is semilocal and \(E(R_R)\) is a DSF-module with small radical, then \(R\) is basic, semiperfect, and right self-injective.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16L30 Noncommutative local and semilocal rings, perfect rings
16L60 Quasi-Frobenius rings
16P20 Artinian rings and modules (associative rings and algebras)
16P40 Noetherian rings and modules (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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References:

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