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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Picard groups and infinite matrix rings


Authors: Gene Abrams and Jeremy Haefner
Journal: Trans. Amer. Math. Soc. 350 (1998), 2737-2752
MSC (1991): Primary 16A42, 16A64
MathSciNet review: 1422591
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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 $R$.


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Additional Information

Gene Abrams
Affiliation: Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933
Email: abrams@math.uccs.edu

Jeremy Haefner
Affiliation: Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933
Email: haefner@math.uccs.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01942-4
PII: S 0002-9947(98)01942-4
Received by editor(s): November 16, 1995
Article copyright: © Copyright 1998 American Mathematical Society