A note on lower nil M-Armendariz rings

TL;DR

This paper investigates properties of lower nil M-Armendariz rings, establishing conditions under which these properties are preserved through quotients and polynomial extensions involving monoids.

Contribution

It provides new results on when lower nil M-Armendariz properties are maintained in ring quotients and polynomial extensions with monoids.

Findings

If R/I is lower nil M-Armendariz, then R is lower nil M-Armendariz.

Under certain conditions, R/N is lower nil M-Armendariz when R is 2-primal and M-Armendariz.

Polynomial extensions R[N] are lower nil M-Armendariz under specific monoid conditions.

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) subset of 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.