Special Properties of Rings of Skew Generalized Power Series

Abstract

Let R be a ring, S a strictly ordered monoid, and ω: S → End(R) a monoid homomorphism. In [30], Marks, Mazurek, and Ziembowski study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. Following [30], we provide various classes of nonreduced (S, ω)-Armendariz rings, and determine radicals of the skew generalized power series ring R[[S ^≤, ω]], in terms of those of an (S, ω)-Armendariz ring R. We also obtain some characterizations for a skew generalized power series ring to be local, semilocal, clean, exchange, uniquely clean, 2-primal, or symmetric.

Key Words:

• Clean ring
• Exchange ring
• Local
• Nil radical
• Prime radical
• 2-primal
• (S, ω)-Armendariz
• Semilocal
• Skew generalized power series ring

2010 Mathematics Subject Classification:

• 16D15
• 16D40
• 16D70

ACKNOWLEDGMENT

The authors would like to express their deep gratitude to the referee for a very careful reading of the article, and many valuable comments, which have greatly improved the presentation of the article.

Notes

Communicated by A. Smoktunowicz.