$\Delta$-closures of ideals and rings

Abstract:

It is shown that if $R$ is a commutative ring with identity and $\Delta$ is a multiplicatively closed set of finitely generated nonzero ideals of $R$, then the operation $I \to {I_\Delta } = { \cup _{K \in \Delta }}(IK:K)$ is a closure operation on the set of ideals $I$ of $R$ that satisfies a partial cancellation law, and it is a prime operation if and only if $R$ is $\Delta$-closed. Also, if none of the ideals in $\Delta$ is contained in a minimal prime ideal, then ${I_\Delta } \subseteq {I_a}$, the integral closure of $I$ in $R$, and if $\Delta$ is the set of all such finitely generated ideals and $I$ contains an ideal in $\Delta$, then ${I_\Delta } = {I_a}$. Further, $R$ has a natural $\Delta$-closure ${R^\Delta },A \to {A^\Delta }$ is a closure operation on a large set of rings $A$ that contain $R$ as a subring, $A \to {A^\Delta }$ behaves nicely under certain types of ring extension, and every integral extension overring of $R$ is ${R^\Delta }$ for an appropriate set $\Delta$. Finally, if $R$ is Noetherian, then the associated primes of ${I_\Delta }$ are also associated primes of ${I_\Delta }K$ and ${(IK)_\Delta }$ for all $K \in \Delta$.