Assassins and torsion functors. II (Q2154562)

Let \(R\) be commutative ring with identity, \(\mathfrak{a}\) an ideal of \(R\) and \(M\) an \(R\)-module. Recall that the \(\mathfrak{a}\)-torsion functor is a subfunctor of the identity functor, on the category of \(R\)-modules and \(R\)-homomorphisms, that defined by \[\Gamma_{\mathfrak{a}}(M):=\{x\in M\mid \exists n\in \mathbb{N} \text{ such that } \mathfrak{a}^n\subseteq (0:_Rx)\}.\] Also, recall that the set of associated primes and weakly associated primes of \(M\) is defined, respectively, by \[\text{Ass}_RM:=\{\mathfrak{p}\in \text{Spec} R\mid \exists x\in M\text{ such that } \mathfrak{p}=(0:_Rx)\}\] and \[\text{Ass}_R^fM:=\{\mathfrak{p}\in \text{Spec} R\mid \exists x\in M \text{ such that } \mathfrak{p} \text{ is minimal} \text{ over } (0:_Rx)\}.\] If \(R\) is Noetherian, it is known that \begin{itemize} \item[(i)] \(\text{Ass}_R^fM=\text{Ass}_RM\). \item[(ii)] \(\text{Ass}_R(\Gamma_{\mathfrak{a}}(M))=\text{Ass}_RM\cap \text{V}(\mathfrak{a})\). \item[(iii)] \(\text{Ass}_R(M/\Gamma_{\mathfrak{a}}(M))=\text{Ass}_RM\setminus \text{V}(\mathfrak{a})\). \end{itemize} In this paper, a new subfunctor of the identity functor is introduced, called large \(\mathfrak{a}\)-torsion functor, which is defined by \[\overline{\Gamma}_{\mathfrak{a}}(M):=\{x\in M\mid\mathfrak{a}\subseteq \sqrt{(0:_Rx )}\}.\] Clearly, \(\overline{\Gamma}_{\mathfrak{a}}(M)=\Gamma_{\mathfrak{a}}(M)\) when \(R\) is Noetherian. The author investigates the ideals \(\mathfrak{a}\) and the \(R\)-modules \(M\) for which the following equalities hold: \begin{itemize} \item[(i)] \(\text{Ass}_R^f(\Gamma_{\mathfrak{a}}(M))=\text{Ass}_R^fM\cap \text{V}(\mathfrak{a})\). \item[(ii)] \(\text{Ass}_R^f(\overline{\Gamma}_{\mathfrak{a}}(M))=\text{Ass}_R^fM\cap \text{V}(\mathfrak{a})\). \item[(iii)] \(\text{Ass}_R(M/\Gamma_{\mathfrak{a}}(M))=\text{Ass}_RM\setminus \text{V}(\mathfrak{a})\) \item[(iv)] \(\text{Ass}_R^f(M/\Gamma_{\mathfrak{a}}(M))=\text{Ass}_R^fM\setminus \text{V}(\mathfrak{a})\). \item[(v)] \(\text{Ass}_R(M/\overline{\Gamma}_{\mathfrak{a}}(M))=\text{Ass}_RM\setminus \text{V}(\mathfrak{a})\) \item[(vi)] \(\text{Ass}_R^f(M/\overline{\Gamma}_{\mathfrak{a}}(M))=\text{Ass}_R^fM\setminus \text{V}(\mathfrak{a})\). \end{itemize}