PICARD-GROUPS AND INFINITE MATRIX-RINGS

We describe a connection between the Picard group of a ring with local units T and the Picard group of the unital overring End(T-T). Using t his connection, we show that the three groups Pic(R), Pic(FM(R)), and Pic(RFM(R)) are isomorphic for any unital ring R. Furthermore, each el ement of Pic(RFM(R)) arises from an automorphism of RFM(R), which yiel ds an isomorphsm between Pic(RFM(R)) and Out(RFM(R)). As one applicati on we extend a classical result of Rosenberg and Zelinsky by showing t hat the group Out(R)(RFM(R)) is abelian for any commutative unital rin g R.