Ore extensions of Jacobson rings.
For a Jacobson ring \(R\), it is well-known that \(R[x]\) is a Jacobson ring, but this is not necessarily true for the skew polynomial extension \(R[x;\alpha]\). For an automorphism \(\alpha\) conditions are known when it will be the case. In this paper the authors consider this problem and add to the complexity by requiring that \(\alpha\) is only a monomorphism and not necessarily surjective.NEWLINENEWLINE Fix a monomorphism \(\alpha\colon R\to R\). An ideal \(I\) of \(R\) is called strongly \(\alpha\)-prime if \(\alpha^{-1}(I)=I\) and for all ideals \(B\) and \(C\) of \(R\) with \(\alpha(C)\subseteq C\) and \(BC\subseteq I\), it follows that \(B\subseteq I\) or \(C\subseteq I\). The ring \(R\) is called strongly \(\alpha\)-prime if \(\{0\}\) is a strongly \(\alpha\)-prime ideal of \(R\).NEWLINENEWLINE A ring \(R\) may satisfy the following conditions:NEWLINENEWLINE \((A_1)\) For each strongly \(\alpha\)-prime ideal \(I\) of \(R\), \((R/I)[x;\alpha]\) is semiprimitive.NEWLINENEWLINE \((A_2)\) For each strongly \(\alpha\)-prime ideal \(I\) of \(R\), \(R/I\) is semiprime.NEWLINENEWLINE \((A_3)\) For each prime ideal \(P\) of \(R[x;\alpha]\) with \(x\notin P\), \(x+P\) is a regular element of \(R[x;\alpha]/P\).NEWLINENEWLINE The main result shows that if a Jacobson ring \(R\) satisfies these three conditions, then \(R[x;\alpha]\) is a Jacobson ring. This result covers the known cases for this conclusion. It is also shown that any left Noetherian ring fulfills these three conditions.
scientific article; zbMATH DE number 6339512
scientific article