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STABLE RANK OF CORNER RINGS P. Ara and K. R. Goodearl
 

Summary: STABLE RANK OF CORNER RINGS
P. Ara and K. R. Goodearl
Abstract. B. Blackadar recently proved that any full corner pAp in a unital C*­algebra
A has K­theoretic stable rank greater than or equal to the stable rank of A. (Here p is a
projection in A, and fullness means that ApA = A.) This result is extended to arbitrary
(unital) rings A in the present paper: If p is a full idempotent in A, then sr(pAp) # sr(A).
The proofs rely partly on algebraic analogs of Blackadar's methods, and partly on a new
technique for reducing problems of higher stable rank to a concept of stable rank one for skew
(rectangular) corners pAq. The main result yields estimates relating stable ranks of Morita
equivalent rings. In particular, if B # = EndA (P ) where PA is a finitely generated projective
generator, and P can be generated by n elements, then sr(A) # n· sr(B) - n + 1.
Introduction
The theory of stable range of rings was developed by H. Bass [2] and L. N. Vaserstein
[10]. As is now common, we define the stable rank of a ring A, denoted sr(A), to be the
least positive integer n such that A satisfies Bass's n­th stable range condition, or # if no
such n exists. It is well known that stable rank is not Morita invariant. In fact, Vaserstein
[10] computed the stable rank of a matrix ring M n (A), obtaining the following amazing
formula
sr(M n (A)) = # sr(A) - 1
n

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics