Categorical equivalences and realization theorems
The main result of this paper is the following: Let \({\mathcal C}\) be a Grothendieck category having a set \((U_i)_{i\in I}\) of small projective generators, let \(A^0\) be a certain unital subring of the ring of endomorphisms of \(\prod_{i\in I} U_i\), and \(I\) a certain idempotent left ideal of \(A^0\). Then, for any subring \(A\) (not necessarily unital) of \(A^0\) containing \(I\), the category \({\mathcal C}\) is equivalent to the category \(\text{Gen}(_AI)\) of all unital left \(A\)-modules generated by \(I\). This theorem allows to provide a unified approach to various results concerning realization theorems of categories of graded objects as module categories, obtained by Cohen \& Montgomery, Beattie, Albu \& Năstăsescu, Albu and del Rio. The authors apply the theorem in the next section to the so-called ``rings graded by categories'', which include group-graded rings.
scientific article; zbMATH DE number 952351