Abstract
Rege and Chhawchharia, as well as Nielsen, introduced the concept of a right McCoy ring, based on McCoy's theorem in 1942 for the annihilators in polynomial rings over commutative rings. Alhevaz et al. in 2012 investigated a natural generalization of a right McCoy ring that is called a weak $(\sigma, \delta)$-skew McCoy ring, where $\sigma$ is an endomorphism and $\delta$ is a $\sigma$-derivation of a given ring. In this paper, we continue to investigate the structural properties of these rings and examine their behavior under various extensions. Specifically, we prove that while the full matrix ring over a weak $(\sigma, \delta)$-skew McCoy ring does not generally preserve this property, the upper triangular matrix ring does under suitable conditions. Furthermore, we explore the stability of this property in polynomial extensions, direct products, classical right quotient rings, and skew power series rings.