A note on $\sigma(*)$-rings and their extensions

Let $R$ be an associative ring with identity $1\neq0$, and $\sigma$ an endomorphism of $R$. We recall $\sigma(*)$ property on $R$ (i.e. $a\sigma(a)\in P(R)$ implies $a\in P(R)$ for $a\in R$, where $P(R)$ is the prime radical of $R$). Also recall that a ring $R$ is said to be $2$-primal if and only if the prime radical $P(R)$ and nil radical are same, i.e. if the prime radical is a completely semiprime ideal. It can be seen that a $\sigma(*)$ ring is a $2$-primal ring.
Let $R$ be a ring and $\sigma$ an automorphism of $R$. Then we know that $\sigma$ can be extended to an automorphism of the skew polynomial ring $R[x;\sigma]$. In this paper we show that if $R$ is a Noetherian ring and $\sigma$ is an automorphism of $R$ such that $R$ is a $\sigma(*)$-ring, then $R[x;\sigma]$ is also a $\sigma(*)$-ring.