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ABSTRACT
For a monoid M, we introduce M-Armendariz rings, which are generalizations of Armendariz rings; and we investigate their properties. Every reduced ring is M-Armendariz for any unique product monoid M. We show that if R is a reduced and M-Armendariz ring, then R is M × N-Armendariz, where N is a unique product monoid. It is also 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-Armendariz. Moreover, we study the relationship between the Baerness and the PP-property of a ring R and those of the monoid ring R[M] in case R is M-Armendariz.
ACKNOWLEDGMENT
The author wishes to express his sincere thanks to the referee for his or her valuable suggestions, which simplified the proof of Proposition 1.1 and Corollary 1.2 and led to the new version of Proposition 1.1.
Supported by the National Natural Science Foundation of China (10171082), TRAPOYT and NWNU-KJCXGC212.