Semiprimeness, quasi-Baerness and prime radical of skew generalized power series rings

ABSTRACT

Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.

KEYWORDS:

• Prime radical
• Quasi-Baer ring
• (S,ω)-Baer ring
• (S,ω)-prime
• (S,ω)-quasi Baer ring
• (S,ω)-semiprime
• skew generalized power series ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S99
• 16W60
• Secondary: 06F05
• 16P60
• 16S36
• 16U80
• 20M25

Acknowledgments

The authors wish to express their sincere thanks to Professor A. Smoktunowicz and to the referee for a very careful reading of the paper and valuable comments which have definitely improved the paper.