On some classes of semi-Artinian rings.
As it is admitted, this article is a continuation of the author's work [in [1] = Sib. Mat. Zh. 49, No. 4, 721-738 (2008); translation in Sib. Math. J. 49, No. 4, 575-586 (2008; Zbl 1154.16004); Sib. Mat. Zh. 50, No. 3, 481-488 (2009); translation in Sib. Math. J. 50, No. 3, 379-384 (2009; Zbl 1212.16019)]. A right \(R\)-module \(M\) is called weakly regular if any submodule of \(M\) not in the Jacobson radical of \(M\), has a nonzero direct summand of \(M\). If all right \(R\)-modules are weakly regular there may exist a left \(R\)-module which is not weakly regular, [see [1, loc. cit.] Corollaries 3.8, 3.9, and their remark]. An associative ring \(R\) with \(1\neq 0\), not necessarily commutative, is called a right semiartinian ring (resp. \(V\)-ring) if every non-zero right module over \(R\) has a simple submodule (resp. if the Jacobson radical of every right module over \(R\) is zero). By an \(SV\)-ring one means a ring which is both right and left semiartinian \(V\)-ring. In the main result of this article (Theorem 3.11), which describes the rings over which all modules (right or left) are weakly regular, the author gives seven equivalent conditions for a ring \(R\) to be an \(SV\)-ring. Let us mention some of the interesting ones [which are also in [1, loc. cit.], Theorem 3.7]: \(R\) is an \(SV\)-ring if and only if \(R\) is a regular (in the sense of von Neumann) ring over which all modules (whether right or left) are weakly regular, or if and only if \(R\) is a right \(V\)-ring with the latter property. Concerning the latter two equivalences, it is interesting to recall that in commutative rings, \(V\)-rings and regular rings coincide, whereas in non-commutative rings this coincidence may not occur. A ring \(R\) is called mod-retractable if for any right \(R\)-module \(M\) and any nonzero submodule \(N\) of \(M\), \(\Hom(M,N)\neq 0\). \textit{A. Haghany, M. R. Vedadi} and the reviewer [in Bull. Iran. Math. Soc. 35, No. 2, 37-45 (2009; Zbl 1194.16007)] have shown that every commutative semiartinian ring is mod-retractable. In the article under review the latter result and its converse (Theorem 3.6) is proved for more general rings, namely, if \(R\) is a right duo ring (i.e., every right ideal is 2-sided), then \(R\) is right semiartinian if and only if it is mod-retractable. Some other interesting facts about mod-retractable rings are also observed.
scientific article; zbMATH DE number 6133170
scientific article