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Let $R$ be a ring and $\sigma$ an endomorphism of $R$. We give a generalization of McCoy’s Theorem [ Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957), 28–29] to the setting of skew polynomial rings of the form $R[x;\sigma ]$. As a consequence, we will show some results on semicommutative and $\sigma$-skew McCoy rings. Also, several relations among McCoyness, Nagata extensions and Armendariz rings and modules are studied.