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Modules graded by G-sets. (English) Zbl 0721.16025
Let R be an associative ring with unity graded by a group G. By a G-set we mean a set A with a G-action on A. A left R-module N is said to be graded by A if \(N=\oplus_{x\in A}N_ x\) for some additive subgroups \(N_ x\) such that \(R_ gN_ x\subseteq N_{gx}\) for \(g\in G\), \(x\in A\). The aim of the paper is to study the category (G,A,R)-gr consisting of the left R-modules graded by a G-set A with degree preserving R-linear maps as morphisms. The inspiration comes from the paper of E. Dade [J. Reine Angew. Math. 369, 40-86 (1986; Zbl 0583.16001)], where some special cases of G-set gradations are applied to the Clifford theory of graded rings. In the first part of the paper the authors show that (G,A,R)-gr is a Grothendieck category. Then, the smash product construction R#G for a ring graded by a finite group G is extended to the case R#A of a finite G-set A. It is shown that (G,A,R)-gr is isomorphic to the category R#A-mod. Moreover, a matrix characterization of R#A is found. In the second part of the paper, certain functors introduced by Dade for G-sets of the form G/H (the left cosets of a subgroup H of G with G acting by left translations) are applied to general G-set gradations. These functors are particularly applied to the study of injective objects in (G,A,R)-gr.

16W50 Graded rings and modules (associative rings and algebras)
16D90 Module categories in associative algebras
16S40 Smash products of general Hopf actions
16D50 Injective modules, self-injective associative rings
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