Semisimple Strongly Graded Rings

Let G be a finite group and R a strongly G-graded ring. The question of when R is semisimple (meaning in this paper semisimple artinian) has been studied by several authors. The most classical result is Maschke’s Theorem for group rings. For crossed products over fields there is a satisfactory answer given by Aljadeff and Robinson [3]. Another partial answer for skew group rings was given by Alfaro, Ara and del Ŕıo [1]. A reduction of the problem to crossed products over division rings was first given by Jespers and Okniński [10] and a more constructive version was given by Haefner and del Ŕıo [8]. So, in order to give a complete answer to the problem there is still a gap between crossed products over division rings and crossed products over fields. The first aim of the paper is to fill this gap, showing that the semisimplicity question for crossed products over division rings reduces to the same question for crossed products over fields. In Theorem 3.2 we make this reduction and then in Theorem 3.3 we put together all the pieces of the puzzle. A strongly G-graded ring R with identity component A induces a group homomorphism σ : G → Pic(A) (see Section 2 for the details). As a consequence of Theorem 3.3, one deduces some necessary conditions for a group homomorphism σ : G → Pic(A) to be induced by a semisimple strongly G-graded ring, namely the conditions 1, 2 and 3 of Corollary 3.5. The Twisting Problem asks whether these necessary conditions are also sufficient, that is given a group homomorphism σ : G → Pic(A) satisfying conditions 1, 2 and 3 of Corollary 3.5, is there a semisimple strongly G-graded ring with coefficient ring A inducing σ? This problem has been investigated in [3] and in [4] for (outer) actions on fields. Our second result (Theorem 4.6) shows that the Twisting Problem for G a cyclic group and A finitely generated as a module over its centre has always a positive solution. Our solution to the problem of semisimplicity of strongly graded rings has an application to actions of finite groups on division rings of prime characteristic. We include this in the last section of the paper. We show that if G is a finite group acting on a division ring D of characteristic p and H is the kernel of this action, then trG(D) 6= 0 if and only if the elements of a p-Sylow subgroup of G are linearly independent over D if and only if the elements of a p-Sylow subgroup of H are linearly independent over D.