Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions

Abstract

Let \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\) (\(\mathbb{Q}\) the  field of rational numbers). Let   \(\sigma\)  be  an automorphism  of   R  and  \(\delta\) a \(\sigma\)-derivation of \(R\). Let further \(\sigma\) be such that \(a\sigma(a)\in N(R)\) implies that \(a\in N(R)\) for \(a\in R\), where \(N(R)\) is the set of nilpotent elements of \(R\). In this paper we study the associated prime ideals of Ore extension \(R[x;\sigma,\delta]\) and we prove the following in this direction:

Let \(R\) be a semiprime right Noetherian ring which is also an algebra over \(\mathbb{Q}\). Let \(\sigma\) and \(\delta\) be as above. Then \(P\) is an associated prime ideal of \(R[x;\sigma,\delta]\) (viewed as a right module over itself) if and only if there exists an associated prime ideal \(U\) of \(R\) with \(\sigma(U) = U\) and \(\delta(U)\subseteq U\) and \(P = U[x;\sigma,\delta]\).

We also prove that if \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\), \(\sigma\) and \(\delta\) as usual such that \(\sigma(\delta(a))=\delta(\sigma(a))\) for all \(a\in R\) and \(\sigma(U) = U\) for all associated prime ideals \(U\) of \(R\) (viewed as a right module over itself), then \(P\) is an associated prime ideal of \(R[x;\sigma,\delta]\) (viewed as a right module over itself) if and only if there exists an associated prime ideal \(U\) of \(R\) such that \((P\cap R)[x;\sigma,\delta] = P\) and \(P\cap R = U\).