Ideal multiplication in Artinian rings

Previously, I described the fundamental importance of quantales in commutative algebra, number theory, etc. In particular, while the non-negative integers naturally form a commutative aperiodic quantale, multiplication in its extension structures is generally not aperiodic. We can recover this desired property of aperiodicity for general commutative rings by instead transitioning to ideal multiplication. This makes ideal multiplication an entirely order-theoretic concept.

Recall for example that the meet of two elements in a lattice is defined to be the "greatest lower bound." This very definition admits immediate order-theoretic generalisation as we can consider other lower bounds and partially order them by how great they are. While a basic order-theory considers semilattices, a more advanced theory can produce all kinds of bounds-producing functions naturally associated to partial orders. The previously mentioned multiplication of the natural numbers arises naturally from the order dual of the lattice of set partitions. Ideal multiplication is an example of such a bound-producing function and it produces a less great lower bound then intersection. This can be clarified more, by considering a special case.

Proposition. ideal multiplication in Artinian commutative rings has bounded index

Proof. ideal multiplication is a monotone and anti-extensive operation, therefore if we consider any ideal $I$ an Artinian ring, then its power $I^2$ is going to be less then $I$ and this can be continued to get a descending chain $I \supseteq I_2 \supseteq I_3 \supseteq I_4 ...$ which clearly must terminate after a finite number of steps. Therefore, there exists $n$ such that $I^{n+1} = I^n$ which is the index of the ideal. In other words, the ideal generates a commutative aperiodic monogenic semigroup with $n$ elements.

This naturally leads to the fact that Artinian integral domains are fields, and that Artinian rings are have T1 spectrum. However, the important point right now is that Artinian rings have a special property relating to ideal multiplication. To summarise:

• Ideal multiplication semigroups are always group-free but they are not always of finite index. The quantale $Ideals(\mathbb{Z})$ for example only has finite index for its bounds.
• Ideal multiplication is always of finite index for Artinian rings

In general, that a semigroup is aperiodic means that it is group-free. Ideal multiplication is always group-free, but it isn't always of finite index unless the commutative ring it arises from is Artinian.