Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2015; 30(4): 363-377

Printed October 31, 2015

https://doi.org/10.4134/CKMS.2015.30.4.363

Copyright © The Korean Mathematical Society.

Zero divisor graphs of skew generalized power series rings

Ahmad Moussavi and Kamal Paykan

Tarbiat Modares University, Tarbiat Modares University

Abstract

Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $\omega: S\rightarrow {\rm End}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\omega ]]$ 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 investigate the interplay between the ring-theoretical properties of $R[[S,\omega ]]$ and the graph-theoretical properties of its zero-divisor graph $\overline{\Gamma}(R[[S,\omega ]])$. Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

Keywords: zero-divisor graph, diameter, girth, skew generalized power series ring, skew power series ring, reduced ring

MSC numbers: 16S99, 16W60, 05C12