Duality theorems for rings with actions or coactions.

*(English)*Zbl 0647.16010The 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.

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 |

##### Keywords:

coactions; rings with local units; smash product; duality theorem for group actions; Morita context; strongly graded; group of automorphisms; skew group ring; ring of matrices; group graded rings
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##### References:

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