Attached prime ideals of generalized inverse polynomial modules. (English) Zbl 1384.16037
Summary: Let \(R\) be a ring, \((S,\geq)\) a strictly totally ordered monoid which is also artinian and \(\omega:S \to \mathrm{Aut}(R)\) a monoid homomorphism. Given a right \(R\)-module \(M\), denote by \([M^{S,\leq}]_{[[R^{S,\leq},\omega]]}\) the generalized inverse polynomial module over the skew generalized power series ring \([[R^{S,\leq},\omega]]\). It is shown in this paper that if \(M_R\) is a completely \(\omega\)-compatible module and \(I\) an attached prime ideal of \(M_R\), then \([[I^{S,\leq},\omega]]\) is an attached prime ideal of \([M^{S,\leq}]_{[ [R^{S,\leq},\omega]]}\), and that if \([M^{S,\leq}]_R\) is a completely \(\omega\)-compatible Bass module, then every attached prime ideal of \([M^{S,\leq}]_{[ [R^{S,\leq},\omega]]}\) can be written as the form of \([[I^{S,\leq},\omega]]\) where \(I\) is an attached prime ideal of \(M_R\).
MSC:
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
Keywords:
skew generalized power series ring; generalized inverse polynomial module; attached prime ideal; completely \(\omega\)-compatible moduleReferences:
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