Product of Ideals?

Question

Why does one study product of ideals in a ring? What are the properties of the product of ideals? We know that product of two ideals $I$ and $J$ is contained in $I\cap J.$ What more can we say about the product of two ideals.

Answer 1

Kummer studied products of ideals in so-called Dedekind rings in the context of Fermat's equation $x^n+y^n=z^n$. Here a a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. For Fermat's equation, the ring is $\mathbb{Z}[\zeta_n]$, where $\zeta_n$ is he primitive $n$-th root of unity. This is a factorial ring for all $n\le 22$, but not in general.

Answer 2

One reason to study ideal products is that they allow us to generalize away from principal ideal rings, in which the product of two ideals doesn't give us any more information than the product of two elements.

The rationale for using them to study Dedekind rings has already been mentioned, so I'll go on by adding that the product of ideals allows us a very good generalization of prime ideals and semiprime ideals to noncommutative rings. A proper ideal $P\lhd R$ is said to be prime if for any ideals $A, B\lhd R$, we have $AB\subseteq P\implies A\subseteq P$ or $B\subseteq P$. This definition of 'prime ideal' is more useful than simply adopting the element-wise defintion of commutative algebra verbatim.

I mentioned in a comment that the set of ideals of any ring, using operations of ideal addition and multiplication, has the structure of a semiring; in fact, it is even an idempotent semiring. It is an interesting interpretation, but I'm not sure if anyone has really made any practical use of it.