Lagmental Vicfred

Integrality Is Equivalent to a Finite Stable Module by Vicfred

tags: mathematics, integral extensions, normalization, spectra

An element is integral exactly when the subalgebra it generates is finite as a module. I want the notation, the mechanism, and the failure mode visible at the same time.

Definitions first

An element \(b\) is integral over \(A\) when it satisfies a monic polynomial with coefficients in \(A\). An extension \(A\subseteq B\) is integral when every \(b\in B\) has this property.

$$ b^n+a_{n-1}b^{n-1}+\cdots+a_0=0 $$

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.

$$ b\ \text{integral over }A\Longleftrightarrow A[b]\ \text{is a finite }A\text{-module} $$

A small case in full

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ A[b]=A\langle1,b,\ldots,b^{n-1}\rangle,\qquad b^n=-\sum_{i=0}^{n-1}a_ib^i $$

The reusable statement

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad b^n+a_{n-1}b^{n-1}+\cdots+a_0=0,\\[5pt] \mathsf{C}\;&:\quad b\ \text{integral over }A\Longleftrightarrow A[b]\ \text{is a finite }A\text{-module}. \end{aligned} $$

A nearby false statement

Algebraic over a fraction field and integral over the base ring are different conditions. Denominators make many algebraic elements nonintegral.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] b\ \text{integral over }A\Longleftrightarrow A[b]\ \text{is a finite }A\text{-module} \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Mon 01 September 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.