Some characterizations of 2-primal skew generalized power series rings

Abstract

A skew generalized power series ring $R[[S,\omega ]]$ consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev-Neumann series rings, the “untwisted” versions of all of these, and generalized power series rings. In this paper we obtain necessary and sufficient conditions on R, S, and ω such that the skew generalized power series ring $R[[S,\omega ]]$ is a 2-primal ring. In particular, it is proved that, under suitable conditions, $R[[S,\omega ]]$ is 2-primal if and only if for every minimal prime ideal ${\mathcal{P}}^{*}$ of $R[[S,\omega ]]$ there exists a minimal prime ideal P of R such that P is completely prime and ${\mathcal{P}}^{*}=P[[S,\omega ]].$ Examples to illustrate and delimit the theory are provided.

Communicated by Jose Luis Gomez Pardo

Keywords:

• Skew generalized power series ring
• 2-primal
• minimal prime ideal
• (S, ω)-prime
• prime radical

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S99
• 16W60
• 16S36
• 16N40

Acknowledgments

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.

Additional information

Funding

This research was supported by the Iran National Science Foundation: INSF (No: 95004390).