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Group graded rings and smash products. (English) Zbl 0602.16002

Let \(G\) be a finite group with identity element \(e\). Let \(R\) be a \(G\)-graded algebra over a commutative ring \(k\). As in a paper by M. Cohen and S. Montgomery [Trans. Am. Math. Soc. 282, 237–258 (1984; Zbl 0533.16001)], one forms the smash product \(R\#(kG)^*\) (where \((kG)^*\) is the dual of the group ring \(kG)\), and deduces from certain duality theorems, various connections between \(R\), \(R_e\) and \(R\#(kG)^*\). In this paper the authors characterize \(R\#(kG)^*\) via endomorphism rings of an \(R\)-generator. They deduce from it a slight generalization of the duality theorems proved by Cohen and Montgomery.

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W20 Automorphisms and endomorphisms
16S34 Group rings
16D90 Module categories in associative algebras

Citations:

Zbl 0533.16001
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Full Text: Numdam

References:

[1] M. Cohen - S. Montgomery , Group-Graded Rings, Smash Product and Group Actions , Trans. Am. Math. Soc. , 282 , no. 1 ( 1984 ), pp. 237 - 258 . MR 728711 | Zbl 0533.16001 · Zbl 0533.16001 · doi:10.2307/1999586
[2] C. N - F. Van Oystaeyen , Graded Ring Theory , North-Holland, Math. Library , Vol. 28 ( 1982 ). MR 676974 | Zbl 0494.16001 · Zbl 0494.16001
[3] Bo Stenström , Rings of Quotients , Springer-Verlag , Berlin , 1975 . MR 389953 | Zbl 0296.16001 · Zbl 0296.16001
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