Alahmadi, Adel N. PI-rings with Artinian proper cyclics are Noetherian. (English) Zbl 1301.16023 Int. Electron. J. Algebra 13, 40-42 (2013). Let \(R\) be an associative ring. The main result of this paper asserts that if \(R\) is a PI integral domain in which every proper cyclic right \(R\)-module is Artinian, then \(R\) is a right Noetherian ring. The other results are for group algebras. Let \(KG\) be a group algebra of a solvable group \(G\). If each proper cyclic right \(KG\)-submodule of \(KG\) is Artinian, then \(KG\) is Noetherian. Moreover, let \(G\) be an arbitrary group. If each proper cyclic right \(KG\)-submodule of \(KG\) is Artinian and \(K\)-finite dimensional, then \(KG\) is also Noetherian. The author does not discuss the conditions under which each proper cyclic right \(KG\)-submodule of \(KG\) is Artinian. Reviewer: S. V. Mihovski (Plovdiv) MSC: 16P20 Artinian rings and modules (associative rings and algebras) 16P40 Noetherian rings and modules (associative rings and algebras) 16R10 \(T\)-ideals, identities, varieties of associative rings and algebras 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) Keywords:cyclic right modules; group algebras of solvable groups; right Noetherian rings; right Artinian rings PDFBibTeX XMLCite \textit{A. N. Alahmadi}, Int. Electron. J. Algebra 13, 40--42 (2013; Zbl 1301.16023) Full Text: Link