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.
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