McCoy Rings Relative to a Monoid

Abstract

For a monoid M, we introduce M-McCoy rings, which are a generalization of McCoy rings and M-Armendariz rings; and investigate their properties. We first show that all reversible rings are right M-McCoy, where M is a u.p.-monoid. We also show that all right duo rings are right M-McCoy, where M is a strictly totally ordered monoid. Then we show that semicommutative rings and 2-primal rings do have a property close to the M-McCoy condition. Moreover, it is shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-McCoy. Consequently, several known results on right McCoy rings are extended to a general setting.

Key Words:

• Armendariz rings
• M-Armendariz rings
• McCoy rings
• Semi-commutative rings

2000 Mathematics Subject Classification:

• Primary 16U99
• Secondary 16S15

ACKNOWLEDGMENT

The author thanks the referee for his/her valuable comments and suggestions. This research is supported by the Shahrood University of Technology in Iran.

Notes

Communicated by T. Albu.