Lagmental Vicfred

Zero Divisors on a Finite Module Lie in Associated Primes by Vicfred

Last updated: Wed 22 July 2026

tags: mathematics, primary decomposition, associated primes, support

Over a Noetherian ring, the zero divisors acting on M form the union of its associated primes. I want the notation, the mechanism, and the failure mode visible at the same time.

Set-up

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.

$$ Z_A(M)=\{a\in A:\exists\,0\ne m\in M,\ am=0\} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ Z_A(M)=\bigcup_{\mathfrak p\in\operatorname{Ass}M}\mathfrak p $$

The calculation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{aligned}M=A/(xy),\quad\operatorname{Ass}M&=\{(x),(y)\},\\Z_A(M)&=(x)\cup(y).\end{aligned} $$

What survives abstraction

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad Z_A(M)=\{a\in A:\exists\,0\ne m\in M,\ am=0\},\\[5pt] \mathsf{C}\;&:\quad Z_A(M)=\bigcup_{\mathfrak p\in\operatorname{Ass}M}\mathfrak p. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] Z_A(M)=\bigcup_{\mathfrak p\in\operatorname{Ass}M}\mathfrak p \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Thu 07 May 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.