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Duality theorems for rings with actions or coactions. (English) Zbl 0647.16010
The aim of this paper is to extend the duality theorems of Cohen and Montgomery for rings with actions or coactions to the case of infinite groups, by using the Morita theory for rings with local units.
Let G be an arbitrary group, and R a G-graded ring with identity. Denote by G*R the subring \(\oplus Rp(x)\) of the smash product \(\tilde R{\#}G\) of Quinn; if G is finite, then \(G*R=\tilde R\#G\), and if G is infinite, then G*R is a ring without identity element, but a ring with local units. In order to prove the duality theorem for group actions, the author defines a Morita context for the ring R, which is strict iff R is strongly graded. For a ring S on which G acts as a group of automorphisms, denote by \(S*G\) the skew group ring. Then, from the Morita context defined one gets \(G*(S*G)\cong M_ G(S)^{fin}\), where \(M_ G(S)^{fin}\) is the ring of matrices with entries in S having rows and columns indexed by G and with only finitely many nonzero entries. The duality theorem for group graded rings: \((G*R)*G\cong M_ G(R)^{fin}\) is obtained by using another Morita context.
Reviewer: T.Albu

MSC:
16S34 Group rings
16W50 Graded rings and modules (associative rings and algebras)
16W20 Automorphisms and endomorphisms
16S50 Endomorphism rings; matrix rings
16D90 Module categories in associative algebras
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