Strongly graded rings with the bounded splitting property (Q1364257)

A left module over an associative ring \(R\) has bounded order if it is isomorphic to a submodule of a factor module of a direct sum of copies of \(R/I\) for \(I\) some essential left ideal of \(R\). The ring \(R\) is said to have the bounded splitting property (BSP) if for every left \(R\)-module \(M\) such that its singular submodule \(Z(M)\) has bounded order then \(Z(M)\) is a direct summand of \(M\). The paper under review is mainly devoted to study the BSP for group rings. It is proved that if \(R[G]\) has BSP then either \(G\) is a finite group or \(R\) is isomorphic to a finite direct product \(M_{n_1}(D_1)\times\cdots\times M_{n_t}(D_t)\) of full matrix rings, where each \(D_i\) is a division ring with center \(C_i\) such that \(C_i[G]\) has BSP (Proposition 3.3). Then the authors concentrate their efforts on \(K[G]\) for \(G\) an infinite abelian group and \(K\) a field. They obtain a nice characterization of the BSP for \(K[G]\) combining their theory with results by \textit{K. R. Goodearl} [Mem. Am. Math. Soc. 124 (1972; Zbl 0242.16018)] and \textit{J. M. Goursaud} and \textit{J. Valette} [J. Algebra 34, 205-212 (1975; Zbl 0303.20006)] (Theorem 3.7). Some of this research on the BSP for group rings is based upon the notion of separable functor introduced by \textit{C. Năstăsescu}, \textit{M. Van den Bergh} and \textit{F. Van Oystaeyen} [J. Algebra 123, No. 2, 397-413 (1989; Zbl 0673.16026)]. Section 2 contains some results about the transfer of the BSP between the Grotendieck categories associated to a group-graded ring, by using separable functors.