I am working on a number theory project, and at one stage, I encounter a commutative algebra problem. Vaguely speaking, my hope is to show that two ideals are equal. Now I shall explain the data I am working with:

Let $(R,\mathfrak{m})$ be a commutative, Noetherian local ring of Krull dimension 1, which is also a $\mathbb{Z}_p$-algebra of characteristic 0. For example, such a ring can be $\mathbb{Z}_p[G]$, where $G$ is a finite abelian $p$-group, but I am also interested in more general cases.

Let $A$ be a $\mathfrak{m}$-primary ideal of $R$ such that one can find a generating set consisting of non-zerodivisors. Let $I,J$ be two ideals such that $I\subseteq J$. Assume that we can prove $I\cdot A=J\cdot A$, where $\cdot$ denotes the product of two ideals.

The question is: under what conditions can we conclude that $I=J$? Of course, the question not difficult when the ideal $A$ is principal, with a non-zero-divisor generator.

I am also interested in the different setting where $(R,\mathfrak{m})$ is a Noetherian integral domain of Krull dimension $d\geqslant2$, and $A$ is a prime ideal of height $d-1$.

First of all, via the usual "Determinant Trick", we can show that $J\subset\overline{I}$, where $\overline{I}$ denotes the integral closure of $I$ in the ring $R$. However, I am not able to prove that the ideal $I$ is integrally closed, and it might be false for the ideals that I am interested in.

Secondly, I read some papers on the theory of "cancellation ideals", but being a cancellation ideal is a very strong peoperty, such that my ideal $A$ is not likely to satisfy.

I learnt from the book Integral Closure of Ideals, Rings, and Modules, that this kind of data is called "$I$ is a reduction of $J$ with respect to $A$" (Definition 17.1.3), so I am wondering if there is any reference which develops tools to deal with such data.

I would appreciate it very much if anyone can provide me with some insights. Thank you in advance!