Infinite group-graded rings, rings of endomorphisms, and localization
Let \(R\) be a \(G\)-graded ring and \(U=\oplus_{x\in G} R(x)\), the canonical generator of the category \(R-gr\) of graded left \(R\)-modules. Then \(\text{End}_{R-gr}(U)\) is isomorphic to \(R\nabla G\), the ring of row finite \(G\times G\) matrices with entries from \(R\) such that the \(x-y\) entry is from \(R_{xy^{-1}}\). In Section 2, it is shown that if \(B\) is a subring of \(R\nabla G\) containing the smash product \(\tilde R\#G\) of \textit{D. Quinn} [Trans. Am. Math. Soc. 292, 155-167(1985; Zbl 0584.16001)], then \(R-gr\) is equivalent to a quotient category of \(B\)-mod and \(R\nabla G\) is isomorphic to a ring of quotients of \(B\). In Section 3, for \(M\) a \(G\)-graded \(R\)-module, the graded endomorphism ring \(\text{END}_ R(U(M))\) is studied for \(M\) finitely generated or \(G\) finite. (\(U(M)=\oplus_{x\in G}M(x)\).) It is shown that \(END_ R(U(M))\) is isomorphicto the skew group ring \(\text{End}_{R-gr}(U(M))*G\). The last section shows that \(R\)-mod is equivalent to a quotient category of \(\text{END}_ R(U)\)-mod and \(\text{End}_ R(U)\) is isomorphic to a ring of quotients of \(\text{END}_ R(U)\).
scientific article; zbMATH DE number 4107018
scientific article