On Von Neumann Regular Rings of Skew Generalized Power Series

Abstract

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S∖{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].

Key Words:

• Artinian and narrow set
• Semisimple ring
• Skew generalized power series ring
• Strictly ordered monoid
• Von Neumann regular ring

2000 Mathematics Subject Classification:

• Primary 16E50, 16S99
• Secondary 06F05

ACKNOWLEDGMENT

The first author was supported by Polish KBN Grant 1 P03A 032 27.

Notes

Communicated by E. R. Puczylowski.