Associated primes over Ore extensions.
This paper is concerned with the determination of the associated primes of induced modules \(M[x;\sigma]\cong M\otimes_RS\) over an Ore extension (skew polynomial ring) \(S=R[x;\sigma]\), where \(\sigma\) is only assumed to be an endomorphism of the ring \(R\). In the case that \(M\) satisfies a \(\sigma\)-compatibility condition, namely that \(\sigma^{-1}(\text{ann}_R(m))=\text{ann}_R(m)\) for all \(m\in M\), \textit{S. Annin} showed that the associated primes of \(M[x;\sigma]\) are precisely the ideals induced from associated primes of \(M\) [Commun. Algebra 30, No. 5, 2511-2528 (2002; Zbl 1010.16025); Erratum ibid. 30, No. 10, 5109-5110 (2002)]. Here, by restricting to the case that \(\sigma\) is surjective, the author determines the associated primes for arbitrary induced modules \(M[x;\sigma]\); they are precisely the ideals of \(S\) induced from those ideals of \(R\) (not necessarily prime) which are \(\sigma\)-associated to \(M\) in a certain sense. Specializing to the case that \(R\) is Noetherian (in which case \(\sigma\) must be an automorphism), it is proved that the associated primes of \(M[x;\sigma]\) are precisely the induced ideals \(P_\sigma S\) where \(P\) is an associated prime of \(M\) and \(P_\sigma\) denotes the largest \(\sigma\)-invariant ideal contained in \(P\). Associated primes for induced modules over skew-Laurent extensions \(R[x^{\pm1};\sigma]\), skew power series rings \(R[\![x;\sigma]\!]\), and general Ore extensions \(R[x;\sigma,\delta]\) are studied by \textit{S. Annin} [in op. cit. and J. Algebra Appl. 3, No. 2, 193-205 (2004; Zbl 1060.16029)].
scientific article; zbMATH DE number 2159738