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Group-graded rings and duality. (English) Zbl 0584.16001

In their paper [Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001)] M. Cohen and S. Montgomery have shown that there exists a duality between finite group actions and group gradings on rings. The aim of the paper under review is to give an alternative construction of this duality and to use it to extend some known results of skew group rings to similar results for group-graded rings.
The basic idea of this construction is to define the smash product as a ring of matrices; more precisely, let \(G\) be a finite multiplicative group with \(n\) elements, \(R=\oplus_{x\in G}R_ x\) a \(G\)-graded ring, and \(M_ G(R)\) the ring of \(n\times n\) matrices over \(R\) with the rows and columns indexed by the elements of \(G\). The ring \(R\) can be embedded in \(M_ G(R)\) by a ring monomorphism \(\eta\) sending each \(r=\sum_{x\in G}r_ x\), \((r_ x\in R_ x)\), on the matrix \(\sum_{(x,y)\in G\times G}r_{xy^{-1}}e(x,y)\), where \(e(x,y)\) is the matrix with 1 in the \((x,y)\)-position and zeroes elsewhere. If \(\tilde R\) denotes the image of \(\eta\), then the smash product \(\tilde R\#G\) of \(\tilde R\) with \(G\) is defined as the subring of \(M_ G(R)\) consisting of all matrices \(\alpha\) for which the \((x,y)\)-entry is in \(R_{xy^{-1}}\). This definition agrees with the definition of the smash product given in the above mentioned paper of Cohen and Montgomery.
In the second section of the paper the construction of the smash product is slightly modified to handle infinite groups.
Reviewer: T.Albu

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16S34 Group rings
16D90 Module categories in associative algebras
16N60 Prime and semiprime associative rings

Citations:

Zbl 0533.16001
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References:

[1] M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237 – 258. · Zbl 0533.16001
[2] Miriam Cohen and Louis H. Rowen, Group graded rings, Comm. Algebra 11 (1983), no. 11, 1253 – 1270. · Zbl 0522.16001 · doi:10.1080/00927878308822904
[3] D. S. Passman, It’s essentially Maschke’s theorem, Rocky Mountain J. Math. 13 (1983), no. 1, 37 – 54. · Zbl 0525.16022 · doi:10.1216/RMJ-1983-13-1-37
[4] D. S. Passman, Semiprime crossed products, Houston J. Math. 11 (1985), no. 2, 257 – 267. · Zbl 0569.16009
[5] D. S. Passman, Infinite crossed products and group-graded rings, Trans. Amer. Math. Soc. 284 (1984), no. 2, 707 – 727. · Zbl 0519.16010
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