Combining Local and Von Neumann Regular Rings

All rings R considered are commutative and have an identity element. Contessa called R a VNL-ring if a or 1 a has a Von Neumann inverse whenever a 2 R. Sample results: Every prime ideal of a VNL-ring is contained in a unique maximal ideal. Local and Von Neumann regular rings are VNL and if the product of two rings is VNL, then both are Von Neumann regular, or one is Von Neumann regular and the other is VNL. The ring Zn of integers mod n is VNL iff ðpqÞ j n whenever p and q are distinct primes. The ring R1⁄21⁄2x of formal power series over R is VNL iff R is local. The ring CðXÞ of all continuous real-valued functions on a Tychonoff space X is VNL if and only if at most one point of X fails to be a P-point. All known VNL-rings satisfy SVNL, namely whenever the ideal generated by a (finite) subset of R is all of R; one of its members has a Von Neumann inverse. We show that a ring R is SVNL if and only if all maximal ideals of R are pure except maybe one. We show that Q a2I RðaÞ is an SVNL if and only if there exists a0 2 I; such that Rða0Þ is an SVNL and for all a 2 I fa0g, RðaÞ is a Von Neumann regular ring. Whether every VNL-ring is an SVNL is an open question. Communicated by W. Martindale. *Correspondence: Emad Abu Osba, Department of Mathematics, College of Science, University of Petra, Queen Alia Airport Road, Amman 961343, Jordan; E-mail: emad@ uop.edu.jo. COMMUNICATIONS IN ALGEBRA Vol. 32, No. 7, pp. 2639–2653, 2004