Jain, S. K.; Kanwar, P.; Malik, S.; Srivastava, J. B. \(KD_\infty\) is a CS-algebra. (English) Zbl 0938.16014 Proc. Am. Math. Soc. 128, No. 2, 397-400 (2000). A nonzero submodule \(N\) in a module \(M\) is called essential in \(M\) if every nonzero submodule \(L\) of \(M\) has nonzero intersection with \(N\). A module is said to be CS if every nonzero submodule of \(M\) is essential in a summand of \(M\). A ring is called right CS if its right regular module is CS. Let \(K\) be a field, \(D_\infty\) be the infinite dihedral group and \(C_\infty\) the infinite cyclic group. It is shown that the group algebra \(KD_\infty\) is CS if and only if \(\text{char}(K)\neq 2\). Moreover, the centre \(Z(KD_\infty)\) is a Dedekind domain and \(KD_\infty\), \(KC_\infty\) are CS as \(Z(KD_\infty)\)-modules. Reviewer: Michael Dokuchaev (Sao Paulo) Cited in 3 ReviewsCited in 7 Documents MSC: 16S34 Group rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16D40 Free, projective, and flat modules and ideals in associative algebras 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) Keywords:group algebras; CS-rings; Dedekind domains PDFBibTeX XMLCite \textit{S. K. Jain} et al., Proc. Am. Math. Soc. 128, No. 2, 397--400 (2000; Zbl 0938.16014) Full Text: DOI