Quasi-duo rings and stable range descent. (English) Zbl 1071.16003

Throughout \(R\) is an associative ring with identity and the subrings have an identity possibly different from the identity of the ring. A subring \(S\) of \(R\) is called a corner ring if \(R=S\oplus C\) for an additive subgroup \(C\) of \(R\) such that \(SC\subseteq C\) and \(CS\subseteq C\), in which case \(C\) is called a complement of \(S\) in \(R\). A corner ring \(S\) with identity \(e\) is called a split corner if it has an ideal complement in \(R\) and is called a semisplit corner if it is a split corner in its associated Peirce corner \(eRe\). A ring is called right quasi-duo if every maximal right ideal of \(R\) is an ideal. It is said that \(R\) has stable range at most \(n\) if, for any right unimodular sequence \(r_1,\dots,r_{n+1}\in R\) (that is, the \(r_i\)’s generate \(R\) as a right ideal), there exist \(x_1,\dots,x_n\in R\) such that \(\sum_{i=1}^n(r_i+r_{n+1}x_i)R=R\).
The authors mainly apply the corner ring theory introduced by T. Y. Lam [Corner ring theory: a generalization of Peirce decompositions. I, in: Proceedings of the International Conference on algebras, modules and rings, Lisboa 2003, World Scientific (to appear)] to the study of the stable range of rings. Thus, the main result states that if \(R\) is left and right quasi-duo and has (right) stable range at most \(n\), then any semisplit corner \(S\) of \(R\) also has (right) stable range at most \(n\). But alongside, a variety of results are established and several open questions are raised on right quasi-duo rings.


16E20 Grothendieck groups, \(K\)-theory, etc.
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
19B10 Stable range conditions
16D25 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)
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