On Skew Inverse Laurent-Serieswise Armendariz Rings

Abstract

We study the skew inverse Laurent-serieswise Armendariz (or simply, SIL-Armendariz) condition on R, a generalization of the standard Armendariz condition from polynomials to skew inverse Laurent series. We study relations between the set of annihilators in R and the set of annihilators in R((x ⁻¹; α)). Among applications, we show that a number of interesting properties of a SIL-Armendariz ring R such as the Baer and the α-quasi Baer property transfer to its skew inverse Laurent series extensions R((x ⁻¹; α)) and vice versa. For an α-weakly rigid ring R, R((x ⁻¹; α)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S _ℓ(R) has a generalized countable join in R. Various types of examples of SIL-Armendariz rings is provided.

Key Words:

• Armendariz ring
• Principally quasi-Baer ring
• Skew inverse Laurent series ring
• Quasi-Baer ring

2000 Mathematics Subject Classification:

• 16W60
• 16.55

ACKNOWLEDGMENT

We would like to express deep gratitude to the referee for his/her valuable suggestions which improved the presentation of the article.

Notes

Communicated by V. A. Artamonov.