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\(KD_\infty\) is a CS-algebra. (English) Zbl 0938.16014

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.

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)
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