Combining Local and Von Neumann Regular Rings

Abstract

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 ∈ 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 ℤ _n of integers mod n is VNL iff (pq)² ∤ n whenever p and q are distinct primes. The ring R[[x]] of formal power series over R is VNL iff R is local. The ring C(X) 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 ∏_α∈I R(α) is an SVNL if and only if there exists α₀ ∈ I, such that R(α₀) is an SVNL and for all α ∈ I − {α₀}, R(α) is a Von Neumann regular ring. Whether every VNL-ring is an SVNL is an open question.

Key Words:

• Local rings
• Von Neumann regular rings
• Von Neumann local rings
• Strongly Von Neumann local rings
• P-space
• Essential P-space

Mathematics Subject Classification:

• Primary 16E50, 06E20
• Secondary 54C

Acknowledgments

Notes

^#Communicated by W. Martindale.