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Direct sums of quasi-injective modules, injective covers, and natural classes. (English) Zbl 0802.16007

Let \(R\) be an associative ring with identity and let \(K\) be a natural class, i.e. \(K\) is a collection of unitary right \(R\)-modules such that \(K\) is closed under isomorphisms, submodules, direct sums and injective hulls. Let \(H_ K(R)\) denote the collection of right ideals \(I\) of \(R\) such that \(R/I\in K\). Let (*) denote the following condition: for every ascending chain \(I_ 1\subseteq I_ 2\subseteq I_ 3\subseteq\dots\) with each \(I_ j\) in \(H_ K(R)\), the union \(\bigcup_ j I_ j\in H_ K(R)\). The authors consider the following properties that \(K\) may possess: \(C_ 1(K)\), every direct sum of injective modules in \(K\) is injective; \(C_ 2(K)\), \(H_ K(R)\) has ACC; \(C_ 3(K)\), every direct sum of quasi-injective modules in \(K\) is quasi-injective; \(C_ 4(K)\) every quasi-injective module in \(K\) is injective; \(C_ 5(K)\) every quasi-injective module in \(K\) is \(\Sigma\)-quasi-injective; \(C_ 6(K)\) every \(R\)-module has a \(K\)-injective precover; \(C_ 7(K)\) every \(R\)- module has a \(K\)-injective cover. The following relationships are established between these conditions: \[ C_ 3(K)\Leftrightarrow (C_ 2(K)\text{ and }C_ 4(K))\quad\text{and}\quad C_ 2(K)\Leftrightarrow C_ 5(K)\Leftrightarrow C_ 6(K)\Leftrightarrow C_ 7(K). \] If in addition \(K\) satisfies (*) then (i) \(C_ 4(K)\Rightarrow C_ 2(K)\) and (ii) every module in \(K\) is semisimple \(\Leftrightarrow\) every cyclic module in \(K\) is injective.

MSC:

16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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