Group - Graded Rings and Duality

We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups. Introduction. In the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings. In the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products. I would like to thank my thesis advisor, D. S. Passman, for his guidance and encouragement throughout the writing of this paper. 1. Finite group-graded rings and the smash product. Let G he a multiplicative group. An associative ring with identity is said to be G-graded if (i) r= e E Ri*) xeG is a direct sum of additive subgroups R(x), with R(x)R(y) c R(xy). It follows that necessarily 1„ g R(l) so that each R(x) is a unitary .R(l)-bimodule. If r cz R, we write r(x) for the component of r in R(x) so that r = zZxecr(x). R is said to be strongly G-graded if R(x)R(y) = R(xy) for all x,y cz G. Throughout this section R is assumed to be G-graded, where G is a finite group with \G\ = n. Let MG(R) denote the set of n X n matrices over R with the rows and columns indexed by the elements of G. If a cz MC(R), we write ax y for the entry in the [x, y]-position of a. Then if a, B g MG(R), the matrix product aB is given by (2) («/*)*.,= E«,,Ar zs=G If U C G is any subset of G, let R(U) = LxsUR(x). In particular, R = R(G). Now suppose H cz G is a subgroup of G. We define R (H} c MG(R) by R{H) = { a cz Mc(R)\axy cz R(xHy~1)}, that is, R{ H) = Ex v^cR(xHy~1)e(x, y), where e(x, y) is the matrix unit with 1^ in the [x, y]-position and zeroes elsewhere. Note that R{G} = MC(R). Received by the editors September 27, 1984. 1980 Mathematics Subject Classification. Primary 16A03. ^1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 155 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use