Computing the closure of a support (Q6600757)

Let \(R\) be a commutative ring and \(E\) be an \(R\)-module. Recall that the support of the \(R\)-module \(E\), Supp\(_R(E) \), is equal to \(\{P\in\) Spec\((R) \mid E_P \neq 0\}\) and the (Bourbaki weak) assassinator Ass\(_R(E)\) is the set of prime ideals (associated prime ideals) \(P\) of \(R\) that are minimal in the set of all prime ideals containing the annihilator of some element \(x \in E\). Therefore, \(\mbox{Ass}_R(E) = \bigcup \{ \mbox{Min}((0 : xR)) \mid x \in E\} \subseteq \mbox{Supp}_R(E)\).\N\NIn the present paper, the authors introduce the notion of Oda ideal, which is an ideal linked to the annihilator of the \(R\)-module \(E\), first defined by S. Oda in 2004 for some particular ring extensions. Precisely, they call the Oda ideal \(\mathcal{O}_R(E)\) (or simply \(\mathcal{O}(E)\)) of an \(R\)-module \(E\) the set of all \(a \in R\) such that \(E_a = 0\). Since, it is known that \(E_a = 0\) if and only if Ass\(_R(E) \subseteq V(a)\), then it is easy to check that \(\mathcal{O}(E) = \bigcap \{\sqrt{(0 :_R xR)} \mid \ x \in E\}= \{a \in R \mid\ \forall x \in E, \exists n \in \mathbb N\) such that \( a^n x = 0 \}\). It follows clearly that \(\mathcal{O}(E)\) is a radical ideal containing \((0 :_R E)\) and Nil\((R)\).\N\NIn the present paper, the authors show that the Zariski closure of the support of an \(R\)-module \(E\) is of the form \(V(\mathcal{O}(E))\). They also give an explicit form of \(\mathcal{O}(E),\) namely \(\mathcal{O}(E) = \bigcap\{P\mid P \in\) Supp\(_R(E)\}\), and study its behavior under various operations of algebra. If \(R \subseteq S\) is a ring extension, they define the Oda ideal of this extension, \(\mathcal{O}(R, S)\), precisely as the Oda ideal of the \(R\)-module \(S/R\); so that \(\mathcal{O}(R, S) = \bigcap \{\sqrt{(R : sR)} \mid s \in S\}\). Therefore,\( (R : S) \subseteq \mathcal{O}(R, S) = \{a \in R \mid R_a = S_a\}\). They provide several applications of \(\mathcal{O}(R, S)\) for ring extensions \(R \subseteq S\) whose supports are closed. The case of crucial and critical ideals of ring extensions is also investigated.