On regular power-substitution.
The author introduces a version of the power-substitution condition [\textit{K. R. Goodearl}, Pac. J. Math. 64, 387-411 (1976; Zbl 0308.16016)] restricted to (von Neumann) regular elements in rings, investigates various equivalent conditions, and relates it to corresponding power-cancellation properties. Specifically, a ring \(R\) is defined to satisfy regular power-substitution provided that for any regular element \(x\in R\), there exists an invertible matrix \(U\in\text{GL}_n(R)\) for some \(n\) such that \(xI_n=xUx\). It is proved that the endomorphism ring of a module \(P\) satisfies regular power-substitution if and only if \(P\) satisfies the following internal power-cancellation property: whenever \(P\cong A\oplus B\cong A\oplus C\), there is a positive integer \(n\) such that \(B^n\cong C^n\). A corollary is that if all matrix rings over \(R\) satisfy regular power-substitution, then every finitely generated stably free \(R\)-module \(P\) has a finite power \(P^s\) which is free. (This result is stated for arbitrary stably free modules, but the theorem used to reduce to the finitely generated case is false.) Conditions equivalent to regular power-substitution are given in terms of stable range \(1\) type conditions; pseudo-similarity; Drazin inverses of products; intertwining of idempotents by invertible matrices; unit-regularity and unit \(\pi\)-regularity. It is proved that every directly finite simple ring satisfies regular power-substitution.
scientific article; zbMATH DE number 5647911
scientific article