Isomorphisms of row and column finite matrix rings

Abstract

Abstract. This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring RF M (R) of row finite matrices and the ring RCF M (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 RCF M (R) and RCF M (S) are isomorphic. This resembles the result of V. P. Camillo (1984) for RF M (R). We also show that the Picard groups of RF M (R) and RCF M (R) are isomorphic, even though the rings RF M (R) and RCF M (R) are never Morita equivalent.

The Globalization Problem for Inner Automorphisms and Skolem-Noether Theorems

“…Let α be an R-automorphism of B = RCFM X (A). By [9] α restricts to an automorphism of I = FM X (A). Let α be the restriction of α to an automorphism of I.…”

“…Now assume that α is an R-automorphism of RFM X (A). By [9,Proposition 6], there is an inner automorphism τ of RFM X (A) such that ατ restricts to an automorphism of I. Thus ατ restricts to an automorphism of B.…”

“…Recall from [1] that every automorphism of a ring with local units A extends uniquely to an automorphism of Q(A) and an automorphism of A is inner if and only if it is the restriction of a inner automorphism of Q(A). Moreover in some cases every automomorphism of Q(A) restricts to an automorphism of A; see [9] and [1].…”