Some natural additive decompositions of elements in bimodules
Let \(R\) and \(S\) be unital rings and \(_RM_S\) a unital \(R-S\)-bimodule. The authors define a category \(\mathcal C\) whose objects are the elements of \(M\) and whose morphisms are elements of \(R\times S\), defined for all \(x,\, y\in M\textrm{ by Hom}_\mathcal C(x,y)=\{(r,s)\colon rx=sy\}\), so that for all \(x\in M,\ \textrm{End}_\mathcal C(x)\) is a subring of \(R\times S\). Factoring by the ideal \(\mathcal I(x,y)=\{(r,s)\colon rx=0=ys\}\) yields a category \(\mathcal D\) which is the primary focus of study in this paper. Using a mix of module and category theoretic techniques, the authors show that \(\mathcal D\) is a pre-additive category in which idempotents split. It follows that for all \(x\in {}_RM_S\) there is a 1-1 correspondence between idempotent elements in End\(_{\mathcal D}(x)\) and the set of all pairs \((x_1, x_2)\) such that \(x=x_1\oplus x_2\). Several examples include \(R=M=R \) and \(M\) Artinian with endomorphism ring \(R\), in which case decompositions of 1 into pairwise orthogonal primitive idempotents correspond to indecomposable decompositions of \(R\). The second half of the paper relates these results to various functors between categories whose objects have semi-local endomorphism rings.
scientific article; zbMATH DE number 7389903
scientific article