A note on lower nil M-Armendariz rings

Abstract

In this article, we prove some results for lower nil M -Armendariz ring. Let M be a strictly totally ordered monoid and I be a semicommutative ideal of R. If R I is a lower nil M -Armendariz ring, then R is lower nil M -Armendariz. Similarly, for above M , if I is 2-primal with N * (R) ⊆ I and R/I is M -Armendariz, then R is a lower nil M -Armendariz ring. Further, we observe that if M is a monoid and N a u.p.-monoid where R is a 2-primal M -Armendariz ring, then R[N ] is a lower nil M -Armendariz ring.Mathematics Subject Classification: 16S36; 16N60; 16Y99.Recall that a monoid M is said to be u.p. monoid (unique product monoid) if for any two non empty finite subsets A, B of M, there exists an element g ∈ M uniquely presented in the form of ab where a ∈ A and b ∈ B. The class of unique product monoids is quite large and important. For example, the right or left ordered monoids, torsion free nilpotent groups, submonoids of a free group. unique product groups and u.p. monoids providing zero divisor problem for group ring. The ring theoretical property of unique product monoid has been established by many authors [6,11,16,17] in past few decades. Proposition 2.1. For a unique product monoid M, every 2-primal ring is lower nil M-Armendariz ring.Proof. Let M be a unique product monoid. Let α = a 1 g 1 + a 2 g 2 + . . . + a n g n , β [M]. Then αβ = 0 in R/N * (R)[M]. Since R/N * (R) is reduced, therefore, by Proposition (1.1) of [11], R/N * (R) is M-Armendariz ring. This implies a i b j ∈ N * (R) for each i and j. Thus, R is a lower nil M-Armendariz ring.Corollary 2.1. For any unique product monoid M, semicommutative ring is lower nil M-Armendariz ring.A monoid M equipped with an order ≤ is said to be an ordered monoid if for any r 1 , r 2 , s ∈ M, r 1 ≤ r 2 implies r 1 s ≤ r 2 s and sr 1 ≤ sr 2 . Moreover, if r 1 < r 2 implies r 1 s < r 2 s and sr 1 < sr 2 , then M is said to be strictly totally ordered monoid. Since each strictly totally ordered monoid is a u.p. monoid, hence by Proposition 2.1, we have the following result.Corollary 2.2. Let M be a strictly totally ordered monoid. Then 2-primal rings are lower nil M-Armendariz rings.The following example shows that the condition u.p. monoid in Proposition 2.1 is not superfluous.