Finitely generated projective modules over row and column finite matrix rings (Q1271904)

Let \(R\) be a unital ring. Denote the ring of row finite matrices over \(R\) by \(\text{RFM}(R)\) and the ring of row and column finite matrices over \(R\) by \(\text{RCFM}(R)\). It is known that \(\text{RFM}(R)\) is isomorphic to the endomorphism ring of \(R^{(\mathbb{N})}\), and hence there is a pair of adjoint functors \(\Hom_R(R^{(\mathbb{N})},-)\colon R\text{-mod}\leftrightarrow\text{RFM}(R)\text{-mod} {:}R^{(\mathbb{N})\otimes-}\). However, there is no analogous result when \(\text{RFM}(R)\) is replaced by \(\text{RCFM}(R)\). In this paper, the author pays special attention to the matrix ring \(\text{RCFM}(R)\) and its relations with properties of \(R\). He shows that if \(P\) is a left \(B\)-module, where \(B=\text{RCFM}(R)\), then \(P\) is a progenerator for \(B\) if and only if \(P^n\cong B\) for some \(n\in\mathbb{N}\) and that \(B\) is Morita equivalent to \(S\) if and only if it is isomorphic to \(\mathbb{M}(S)\) for some \(n\in\mathbb{N}\). Also, the classes of semisimple, perfect and semiprimary rings are described in terms of the properties of \(B\). The following result for semiprimary rings is a remarkable result: \(R\) is semiprimary if and only if \(J(B)\) is nilpotent and \(B/J(B)\cong B_1\times\cdots\times B_n\) for rings \(B_i\) such that every finitely generated projective left \(B_i\)-module is either semisimple or isomorphic to \(B_i\).