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.

Siano $I$ un ideale di un anello $R$ e $\sigma$ una congruenza su un semigruppo $S$. Consideriamo l'anello semigruppo $\left(R/I\right)\left(S/\sigma \right)$ come un'immagine omomorfa dell'anello semigruppo $R\left(S\right)$. Questo è fatto in tre passi: prima studiando l'anello semigruppo $R\left(S/\sigma \right)$, poi $\left(R/I\right)\left(S\right)$ e infine combinando i due casi speciali. In ciascun caso, determiniamo l'ideale che è il nucleo dell'omomorfismo in questione. I risultati corrispondenti per le $C$-algebre, dove $C$ è un anello commutativo, possono essere facilmente dedotti. Alcuni raffinamenti, casi speciali...

Let $R$ be a ring. A right $R$-module $M$ is said to be retractable if $𝕋{Hom}_R(M,N)\ne 0$ whenever $N$ is a non-zero submodule of $M$. The goal of this article is to investigate a ring $R$ for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that $\left(1\right)$ The ring ${\prod }_{i\in ℐ}{R}_i$ is right mod-retractable if and only if each $R_i$ is a right mod-retractable ring for each $i\in ℐ$, where $ℐ$ is an arbitrary finite set. $\left(2\right)$ If $R\left[x\right]$ is a mod-retractable ring then $R$ is a mod-retractable ring.

In this note, for a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$ of a ring $R$, the notion of weakened $(\alpha ,\delta )$-skew Armendariz rings is introduced as a generalization of $\alpha$-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha ,\delta )$-skew Armendariz ring if and only if $T_n\left(R\right)$ is weakened $(\overline{\alpha },\overline{\delta })$-skew Armendariz if and only if $R\left[x\right]/\left(x^n\right)$ is weakened $(\overline{\alpha },\overline{\delta })$-skew Armendariz ring for any positive integer $n$.