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In this paper, we first investigate the relationships between the McCoy module and related modules based on their relationships in rings. After that, we improve some properties of McCoy modules and introduce ZPZC modules which extend the notion of McCoy modules. We observe the structure of ZPZC modules providing a number of examples of problems that arise naturally in the process. Finally, answers to some open questions related to the ZPZC condition are provided.

<abstract><p>Let $ S $ be a multiplicative subset of a ring $ R $. A right ideal $ A $ of $ R $ is referred to as <italic>$ S $-principal</italic> if there exist an element $ s \in S $ and a principal right ideal $ aR $ of $ R $ such that $ As \subseteq aR \subseteq A $. A ring is referred to as an $ S $-<italic>principal right ideal ring</italic> ($ S $-PRIR) if every right ideal is $ S $-principal. This paper examines $ S $-PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of $ S $-PRIRs are investigated, and some practical results are proven. A noncommutative $ S $-PRIR that is not a principal right ideal ring is found, and the $ S $-variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for $ S $-PRIRs are proven.</p></abstract>

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