Goldie ranks of 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 continue the study of skew generalized power series ring $R[[S,\omega ]].$ We investigate the problem when a skew generalized power series ring $R[[S,\omega ]]$ has the same Goldie rank as the ring R, and we obtain partial characterizations for it to be serial semiprime. Finally, we will obtain criterion for skew generalized power series rings to be right nonsingular.

Keywords:

• Skew generalized power series ring
• Goldie rank
• nonsingular ring
• serial semiprime ring
• strictly ordered monoid

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16S99
• 16W60
• 16S36
• 16N40

Acknowledgements

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