On Strong $(a)$-Rings

TL;DR

This paper introduces a strong property $(A)$ in ring theory, explores its transfer in various ring extensions, and identifies rings satisfying property $(A)$ but not the strong version.

Contribution

It defines a new strong property $(A)$, studies its behavior in ring extensions, and distinguishes rings with property $(A)$ from those with strong property $(A)$.

Findings

Identified rings satisfying property $(A)$ but not strong property $(A)$

Analyzed transfer of property $(A)$ and strong property $(A)$ in trivial extensions

Provided examples of rings with property $(A)$ but lacking strong property $(A)$

Abstract

In this paper, we introduce a strong property $(A)$ and we study the transfer of property $(A)$ and strong property $(A)$ in trivial ring extensions and amalgamated duplication of a ring along an ideal. We also exhibit a class of rings which satisfy property $(A)$ and do not satisfy strong property $(A)$.