Armendariz ring with weakly semicommutativity

TL;DR

This paper introduces the weak ideal-Armendariz ring, a new class combining Armendariz and weakly semicommutative properties, and explores its properties and generalizations in ring theory.

Contribution

It defines the weak ideal-Armendariz ring, proves its equivalence with polynomial extensions, and introduces its generalization as strongly nil-IFP, expanding the understanding of ring structures.

Findings

R is weak ideal-Armendariz iff R[x] is weak ideal-Armendariz.

Weak ideal-Armendariz generalizes ideal-Armendariz and relates to strongly nil-IFP.

Conditions under which rings and their quotients are strongly nil-IFP.

Abstract

In this article, we introduce the weak ideal-Armendariz ring which combines Armendariz ring and weakly semicommutative properties of rings. In fact, it is a generalisation of an ideal-Armendariz ring. We investigate some properties of weak ideal Armendariz rings and prove that R is a weak ideal-Armendariz ring if and only if R[x] is weak ideal-Armendariz ring. Also, we generalise weak ideal-Armendariz as strongly nil-IFP and a number of properties are discussed which distinguishes it from other existing structures. We prove that if I is a semicommutative ideal of a ring R and R/I is a strongly nil-IFP, then R is strongly nil-IFP. Moreover, if R is 2-primal, then R[x]/<x^{n}> is a strongly nil-IFP.