In abstract algebra, an associated prime of a module $M$ over a ring $R$ is a type of prime ideal of $R$ that arises as an annihilator of a submodule of $M$. The set of associated primes is usually denoted by $\operatorname{Ass}_{R}(M)$.
In commutative algebra, associated primes are linked to the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal $I$ is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with $\operatorname{Ass}_{R}(R/I)$.
The integral closure of an ideal $I$ of a commutative ring $R$, denoted by $\overline {I}$, is the set of all elements $r\in R$ that are integral over $I.$ That is, there exist $a_{i}\in I^{i}$ such that $$r^{n}+a_{1}r^{n-1}+\cdots +a_{n-1}r+a_{n}=0.$$
It is seen that $I\subseteq \overline{I}\subseteq\sqrt{I}$. I want to know what the relation is between $\operatorname{Ass}_R R/I$ and $\operatorname{Ass}_R R/\overline{I}.$