Countably McCoy rings

Year 2022, Volume: 51 Issue: 3, 725 - 736, 01.06.2022

Samir Bouchiba , Abderrazzak Ait Ouahi Youssef Najem

https://doi.org/10.15672/hujms.910906

Abstract

The main goal of this paper is to study the class of countably $\mathcal {A}$-rings (or the countably McCoy rings) introduced by T. Lucas in [The diameter of a zero divisor graph, J. Algebra 301, 174-193, 2006] which turns out to lie properly between the class of $ \mathcal{A}$-rings (or McCoy rings) and the class of total-$\mathcal{A}$-rings. Also, we introduce and investigate the module theoretic version of the countably $\mathcal {A}$-ring notion, namely the countably $\mathcal {A}$-modules. Our focus is shed on the behavior of the countably $\mathcal {A}$-property vis-à-vis the polynomial ring, the power series ring, the idealization and the direct products. Numerous examples are provided to show the limits of the results.

Keywords

countably McCoy rings, countably McCoy modules, Noetherian ring, $\mathcal{A}$-ring, $\mathcal{A}$-module, zero divisor

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