Transparency of Ore extensions over left $\sigma$-$(S,1)$ rings

Let $R$ be a ring and $\sigma$ be an endomorphism of $R$. Recall that a ring $R$ is said to be a left $\sigma$-$(S,1)$ ring if for $a,b\in R$, $ab=0$ implies that $aRb=0$ and $\sigma(a)Rb=0$. In this paper we discuss a stronger type of primary decomposition (known as transparency) of a left $\sigma$-$(S,1)$ ring $R$, and Ore extension $R[x;\sigma]$.