Lagmental Vicfred

Primary Ideals Have One Radical Prime by Vicfred

Last updated: Mon 06 April 2015

tags: mathematics, primary decomposition, associated primes, support

An ideal is primary when a zero divisor modulo it is forced to be nilpotent. The point is to make the formal expression readable enough to audit line by line.

The mathematical object

For a finite \(A\)-module \(M\), the support \(\operatorname{Supp}M\) records primes where localisation is nonzero. Associated primes identify annihilators of individual elements and expose the irreducible pieces of zero divisors.

$$ Q\triangleleft A,\qquad ab\in Q,\ a\notin Q\Longrightarrow b^n\in Q\ \text{for some }n $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ Q\ \text{primary}\Longrightarrow\sqrt Q=\mathfrak p\ \text{prime} $$

One explicit computation

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \sqrt{(x^m)}=(x),\qquad ab\in(x^m),\ a\notin(x^m)\Longrightarrow b^m\in(x^m)\quad\text{in }k[x] $$

Why the identity matters

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad Q\triangleleft A,\qquad ab\in Q,\ a\notin Q\Longrightarrow b^n\in Q\ \text{for some }n,\\[5pt] \mathsf{C}\;&:\quad Q\ \text{primary}\Longrightarrow\sqrt Q=\mathfrak p\ \text{prime}. \end{aligned} $$

Where it can fail

Primary decomposition is not unique term by term. Under Noetherian hypotheses, the isolated associated primes are canonical even when embedded components move.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] Q\ \text{primary}\Longrightarrow\sqrt Q=\mathfrak p\ \text{prime} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sun 29 January 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.