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Our joint work with Kostia Beidar on the separativity problem for regular rings. (English) Zbl 1137.16017

Chebotar, Mikhail (ed.) et al., Rings and nearrings. Proceedings of the international conference of algebra in memory of Kostia Beidar, Tainan, Taiwan, March 6–12, 2005. Berlin: Walter de Gruyter (ISBN 978-3-11-019952-9/hbk). 111-120 (2007).
A ring \(R\) is said to be ‘separative’ if for all finitely generated projective right \(R\)-modules \(A\) and \(B\), the conditions \(A\oplus A\cong A\oplus B\cong B\oplus B\) imply \(A\cong B\). This notion was introduced by P. Ara, K. R. Goodearl, K. C. O’Meara and E. Pardo, [in Isr. J. Math. 105, 105-137 (1998; Zbl 0908.16002)], and has played a unifying role for certain direct sum cancellation problems in ring theory, \(C^*\)-algebras and Abelian group theory.
The fundamental Separativity Problem for (von Neumann) regular rings asks whether such rings are always separative. The paper under review gives a survey of recent results on this problem obtained by the authors and K. Beidar in [K. C. O’Meara and R. M. Raphael, Algebra Univers. 45, No. 4, 383-405 (2001; Zbl 1058.16009); K. I. Beidar, K. C. O’Meara and R. M. Raphael, Commun. Algebra 32, No. 9, 3543-3562 (2004; Zbl 1074.16005)].
As reported in the last section, one of the results in the latter paper gave rise to a new canonical form for matrices, the \(H\)-form, which has been used by K. C. O’Meara and C. Vinsonhaler [in Linear Algebra Appl. 412, No. 1, 39-74 (2006; Zbl 1087.15507)] as a tool for tackling the following problem: Given \(A_1,\dots,A_k\) commuting \(n\times n\) matrices over the complex numbers, can the matrices be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable?
For the entire collection see [Zbl 1113.16001].

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S50 Endomorphism rings; matrix rings
15A21 Canonical forms, reductions, classification
15A09 Theory of matrix inversion and generalized inverses
01A70 Biographies, obituaries, personalia, bibliographies

Biographic References:

Beidar, K. I.
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