Isomorphisms Between Matrix Rings over a Fixed Ring of Scalars
by
Gene Abrams
University of Colorado at Colorado Springs
Coauthors: P. N. Anh

The ring R is said to have type (n, k) in case there is an isomorphism of free modules _RRⁱ =~ _RR^j if and only if i, j >= n and i \equiv j (modk). Clearly for such i, j there is a ring isomorphism between the matrix rings M_i(R) and M_j(R). We provide examples which show that for a ring of type (n, k) there can exist isomorphisms M_i(R) =~ M_i'(R) where i, i' >= n and i \not \equiv i'(modk). We then investigate examples (provided by G. Bergman) of Invariant Basis Number rings R for which there exists an upward-directed set S subset or equal N with the property that M_i(R) =~ M_j(R) if and only if i, j in S. We show that these IBN examples in fact arise as subrings of the original class of non-IBN rings.

Date received: March 14, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-18.