Isomorphisms of row and column finite matrix rings

Abstract

This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring R F M ( R ) RFM(R) of row finite matrices and the ring R C F M ( R ) RCFM(R) of row and column finite matrices. For example, we prove that two rings R R and S S are Morita equivalent if and only if the rings R C F M ( R ) RCFM(R) and R C F M ( S ) RCFM(S) are isomorphic. This resembles the result of V. P. Camillo (1984) for R F M ( R ) RFM(R) . We also show that the Picard groups of R F M ( R ) RFM(R) and R C F M ( R ) RCFM(R) are isomorphic, even though the rings R F M ( R ) RFM(R) and R C F M ( R ) RCFM(R) are never Morita equivalent.