Regular rings with generalized \(s\)-comparability (Q5943058)

The study of (von Neumann) regular rings \(R\) has led to various comparability conditions, under which certain modules are required to embed in direct sums or summands of copies of others. In particular, \textit{D. Handelman} and the reviewer [J. Pure Appl. Algebra 7, 195-216 (1976; Zbl 0321.16009)] introduced the \(s\)-comparability condition (for a positive integer \(s\)), and the reviewer [Von Neumann regular rings, London, Pitman (1970; Zbl 0411.16007)] introduced the general comparability condition. These were merged by \textit{E. Pardo} [Commun. Algebra 24, No. 9, 2915-2929 (1996; Zbl 0859.16001)], who introduced the generalized \(s\)-comparability condition: For any finitely generated projective right \(R\)-modules \(A\) and \(B\), there is a central idempotent \(e\in R\) such that \(Ae\) embeds in \((Be)^s\) while \(B(1-e)\) embeds in \((A(1-e))^s\). (This reduces to \(s\)-comparability when \(0\) and \(1\) are the only central idempotents in \(R\), while the case \(s=1\) yields general comparability.)NEWLINENEWLINENEWLINEHere the author shows that a number of results originally proved assuming either \(s\)-comparability or general comparability still hold under the hypothesis of generalized \(s\)-comparability. For instance, generalized \(s\)-comparability is Morita invariant, and it need only be checked for cyclic projective modules. Assuming generalized \(s\)-comparability, each prime ideal \(P\) of \(R\) is contained in a unique maximal ideal and contains a unique minimal prime ideal, and the lattice of ideals of \(R/P\) is totally ordered.