Semisimple strongly graded rings
Let \(G\) be a finite group with neutral element \(1\). A strongly \(G\)-graded ring \(R=\bigoplus_{g\in G}R_g\) induces a group homomorphism \(\sigma\colon G\to\text{Pic}(A)\), where \(A=R_1\), and \(\text{Pic}(A)\) denotes the Picard group of the ring \(A\). When \(R\) is a crossed product \(R=A*^\beta_tG\), then this homomorphism comes from the action \(\beta\colon G\to\Aut(A)\) via the canonical map \(\Aut(A)\to\text{Pic}(A)\). Recall that in such a case, the product of \(R\) is defined from a right \(A\)-basis of homogeneous units \(\{u_g:g\in G\}\), namely \(au_g=u_ga^{\beta(g)}\), \(u_gu_h=u_{hg}t(g,h)\), where \(t\colon G\times G\to A^*\) is the twisting. If \(A\) is a division ring with center \(K\), and \(H\) is the kernel of the outer action \(\alpha\colon G\to\text{Out}(A)\) induced by \(\beta\), then \(A*_{t'}H\) is a twisted group ring (i.e. with trivial action), where \(t'\) is a twisting obtained after a diagonal change of basis. Moreover, \(K*_{t'}H\) makes now sense, as \(t'(g,h)\in K\) for every \(g,h\in H\). It is proved (Theorem 3.2) that \(R=A*^\beta_tG\) is semisimple if and only if \(A*_{t'}H\) is semisimple if and only if \(K*_{t'}H\) is semisimple. This leads, in conjunction with Morita theory and the work by \textit{E. Aljadeff} and \textit{D. J. S. Robinson} [J. Pure Appl. Algebra 94, No. 1, 1-15 (1994; Zbl 0809.16029)], to several interesting characterizations of strongly \(G\)-graded semisimple rings (Theorem 3.3). As a consequence, the group homomorphism \(\sigma\colon G\to\text{Pic}(A)\) satisfies some necessary conditions to be induced by a strongly graded semisimple ring. The authors study whether these conditions are also sufficient (this is the `twisting problem'), and give a positive answer when \(G\) is cyclic and \(A\) is finitely generated and projective over its center (Theorem 4.6). The paper is completed with an application to actions of finite groups on division rings of prime characteristic (Theorem 5.1). If \(H\) is the kernel of the action of \(G\) over a division ring \(D\) of characteristic \(p\), then the action has non-zero trace if and only if the elements of a Sylow \(p\)-subgroup of \(G\) (resp. of \(H\)) are linearly independent over \(D\).
scientific article; zbMATH DE number 1868024
scientific article