Isomorphisms of row and column finite matrix rings

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