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For a monoid M, we introduce the concept of 3-M-Armendariz rings, which is a generalization of M-Armendariz rings, and investigate its properties. The results prove that the subrings of 3-M-Armendariz rings are 3-M-Armendariz rings. Every ring satisfying condition (P) is3-M-Armendariz for any unique product monoid M. If a ring R is 3-M-Armendariz and sat-isfies condition (P), then S3(R) is 3-M-Armendariz. Sufficient and necessary conditions are given for a ring R to be 3-M-Armendariz. ? 2014 Academic Publications, Ltd.

ON THE EXTENSION OF ARMENDARIZ RINGS RELATIVE TO A MONOID