On Almost Armendariz Rings

TL;DR

This paper introduces the concept of almost Armendariz rings, a generalization of Armendariz rings, and explores their properties, characterizations, and specific classes, including minimal noncommutative examples.

Contribution

It defines almost Armendariz rings, establishes their equivalence with matrix and polynomial extensions, and characterizes minimal noncommutative cases.

Findings

A ring R is almost Armendariz iff T_n(R) is almost Armendariz.

R is almost Armendariz iff R[x] is almost Armendariz.

Every almost Armendariz ring is weak Armendariz, but not vice versa.

Abstract

In this paper, we introduce the notion of almost Armendariz ring which is the generalization of Armendariz ring and discuss some of its properties. We prove that a ring R is almost Armendariz if and only if n X n upper triangular matrix ring T_{n}(R) is almost Armendariz ring. Similarly, If R is almost Armendariz if and only if R[x] is almost Armendariz. It is observed that every almost Armendariz ring is weak Armendariz but converse need not be true. But, if R is semicommutative ring, then weak Armendariz ring is almost Armendariz ring. Moreover, the class of minimal noncommutative almost Armendariz rings is completely determined, up to isomorphism (minimal means having smallest cardinality).