Picard groups and infinite matrix rings
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Abstract:
We describe a connection between the Picard group of a ring with local units $T$ and the Picard group of the unital overring $End(_TT)$. Using this 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 element of $Pic(RFM(R))$ arises from an automorphism of $RFM(R)$, which yields an isomorphsm between $Pic(RFM(R))$ and $Out(RFM(R))$. As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group $Out_R(RFM(R))$ is abelian for any commutative unital ring $R$.References
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Additional Information
- Gene Abrams
- Affiliation: Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933
- MR Author ID: 190273
- Email: abrams@math.uccs.edu
- Jeremy Haefner
- Affiliation: Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933
- Email: haefner@math.uccs.edu
- Received by editor(s): November 16, 1995
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 2737-2752
- MSC (1991): Primary 16A42, 16A64
- DOI: https://doi.org/10.1090/S0002-9947-98-01942-4
- MathSciNet review: 1422591