Hilbert's Basis Theorem Adds One Polynomial Variable

A polynomial ring over a Noetherian ring remains Noetherian. The point is to make the formal expression readable enough to audit line by line.

Notation

A ring $A$ is Noetherian when ascending chains of ideals stabilise. Equivalently, every ideal $I\triangleleft A$ is finitely generated, so finite data controls all later ideal growth.

$$ A\ \text{Noetherian} $$

The formulas should not be merged too early. The datum $\mathsf D$, the conclusion $\mathsf C$, and the bridge $\Longrightarrow$ have three different logical jobs.

$$ A[x]\ \text{Noetherian} $$

Stress the formula

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.

$$ \begin{aligned}L(I)&=\{\text{leading coefficients of }f\in I\}\triangleleft A,\\L(I)&=(a_1,\ldots,a_r)\Longrightarrow I=(f_1,\ldots,f_r,g_1,\ldots,g_s).\end{aligned} $$

Interpretation

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad A\ \text{Noetherian},\\[5pt] \mathsf{C}\;&:\quad A[x]\ \text{Noetherian}. \end{aligned} $$

Limit of the argument

Noetherian does not mean finite, Artinian, or a domain. Each additional adjective imposes a different chain condition or multiplicative property.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A[x]\ \text{Noetherian} \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Sat 18 June 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

Hilbert's Basis Theorem Adds One Polynomial Variable by Vicfred

Notation

Stress the formula

Interpretation

Limit of the argument