Quasi-duo rings and stable range descent.
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 \textit{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.
scientific article; zbMATH DE number 2136955
scientific article