Extension of Almost Armendariz Rings

TL;DR

This paper introduces a new class of rings called $oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$ ext{ extbf{ extit{ extbf{almost Armendariz}}}}$ rings, extending the concept to skew polynomial rings with endomorphisms, and explores their properties and relations.

Contribution

It generalizes almost Armendariz rings to $ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$ ext{ extbf{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$ ext{ extbf{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$oldsymbol{ ext{ extit{ extbf{$ ext{ extbf{ extit{ extbf{skew almost Armendariz}}}}$ rings considering endomorphisms, and establishes their properties and connections to reversible rings.

Findings

Reversible rings with certain endomorphisms have polynomial rings that are skew almost Armendariz.

The paper characterizes conditions under which polynomial rings inherit the skew almost Armendariz property.

Extension of classical almost Armendariz rings to a broader non-commutative setting.

Abstract

A ring $R$ is said to be an almost Armendariz ring if whenever product of two polynomials in $R[x]$ is zero, then product of their coefficients are in $N_{*}(R)$. In this article, for an endomorphism $\alpha$ on $R$, we define an $\alpha$-almost Armendariz ring of $R$ considering the polynomials in skew polynomial ring $R[x; \alpha]$ instead of $R[x]$. It is the generalisation of an almost Armendariz ring [9] and an $\alpha$-Armendariz ring [4]. Moreover, for an endomorphism $\alpha$ of $R$, we define an $\alpha$-skew almost Armendariz ring, and prove that a reversible ring $R$ with certain condition on endomorphism $\alpha$, its polynomial ring $R[x]$ is an $\overline{\alpha}$-skew almost Armendariz ring.