Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a ring with an automorphism $\sigma$. An ideal $I$ of $R$ is {\it$\sigma$-ideal} of $R$ if $\sigma$($I$) = $I$. A proper ideal $P$ of $R$ is {\it$\sigma$-prime ideal\/} of $R$ if $P$ is a $\sigma$-ideal of $R$ and for $\sigma$-ideals $I$ and $J$ of $R$, $IJ$ $\subseteq$ $P$ implies that $I$ $\subseteq$ $P$ or $J$ $\subseteq$ $P$. A proper ideal $Q$ of $R$ is {\it$\sigma$-semiprime ideal\/} of $Q$ if $Q$ is a $\sigma$-ideal and for a $\sigma$-ideal $I$ of $R$, $I^{2}$ $\subseteq$ $Q$ implies that $I$ $\subseteq$ $Q$. The $\sigma$-prime radical is defined by the intersection of all $\sigma$-prime ideals of $R$ and is denoted by $P_{\sigma}(R)$. In this paper, the following results are obtained: (1)~For a principal ideal domain $R$, $P_{\sigma}(R)$ is the smallest $\sigma$-semiprime ideal of $R$; (2)~For any ring $R$ with an automorphism $\sigma$ and for a skew Laurent polynomial ring $R[x, x^{-1}; \sigma]$, the prime radical of $R[x, x^{-1}; \sigma]$ is equal to $P_{\sigma}(R)[x, x^{-1}; \sigma]$.

Keywords: $\sigma$-semiprime ring, $\sigma$-prime ring, $\sigma$-prime radical, skew Laurent polynomial ring

MSC numbers: 16N99