Skew McCoy rings and σ-compatibility

In this paper, we study [Formula: see text]-skew McCoy rings under the [Formula: see text]-compatible or the [Formula: see text]-semicompatible conditions. We show that if [Formula: see text] is a semicommutative right or left artinian ring which is [Formula: see text]-semicompatible with an epimorphism [Formula: see text], then the Jacobson radical [Formula: see text] is [Formula: see text]-skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every [Formula: see text]-compatible right duo ring is [Formula: see text]-skew McCoy and that for [Formula: see text]-compatible regular rings, the notions of the [Formula: see text]-skew McCoy and the right McCoy coincide. In addition, we show that every [Formula: see text]-semicompatible semicommutative ring is linearly [Formula: see text]-skew Camillo and that every matrix ring over a division ring is linearly [Formula: see text]-skew Camillo for any endomorphism [Formula: see text].