Simplicity of partial skew group rings of abelian groups. (Q2925368)

Recently [in Commun. Algebra 42, No. 2, 831-841 (2014; Zbl 1300.16025)] \textit{J. Öinert} has characterized the simple skew group rings of abelian groups. In the present paper the author generalizes some of these results to rings with local units and to partial skew group rings.NEWLINENEWLINE An associative ring \(A\) is a ring with local units if for every finite set \(\{r_1,r_2,\ldots,r_n\}\subseteq A\) there exists an idempotent \(e\in A\) such that \(er_i=r_i=r_ie\) for all \(i=1,2,\ldots,n\). The notion partial skew group ring is defined by \textit{M. Dokuchaev} and \textit{R. Exel} [Trans. Am. Math. Soc. 357, No. 5, 1931-1952 (2005; Zbl 1072.16025)].NEWLINENEWLINE Let \(A\) be an associative ring with local units and let \(G\) be an additive abelian group. Suppose that \(\alpha=\{\alpha_g:D_{-g}\to D_g\mid g\in G\}\) is a partial action of the group \(G\) on the ring \(A\), where \(D_g\) is an ideal of \(A\) and \(\alpha_g\) is an isomorphism for each \(g\in G\). Let \(R=A*_\alpha G\) be the corresponding partial skew group ring of \(G\) over \(A\) with the partial action \(\alpha\). Then every element of \(R\) is a finite sum of the form \(\sum_{g\in G}a_g\delta_g\), where \(\delta_g\) are symbols and \(a_g\in D_g\) for each \(g\in G\). We say that an ideal \(I\subseteq A\) is \(G\)-invariant if \(\alpha_g(I\cap D_{-g})\subseteq I\cap D_g\) for all \(g\in G\). If \(\{0\}\) and \(A\) are the only \(G\)-invariant ideals of \(A\), then \(A\) is \(G\)-simple.NEWLINENEWLINE The main result asserts that \(R\) is a simple ring if and only if \(A\) is \(G\)-simple and the center of \(e\delta_0Re\delta_0\) is a field for each local unit \(e\in A\). The author applies this result to characterize the simplicity of partial skew group rings in two cases arising from partial actions on sets.