r one s two intersection of height one localisations

Given: The objects and hypotheses in the statement, including the Axiom of Choice (The Axiom of Choice); cited dependent-choice and resolution-existence hypotheses are retained.

serre r k and s k conditions: For a commutative Noetherian ring $R$ and an integer $j\ge 0$, condition $(R_j)$ means that $R_{\mathfrak{p}}$ is regular whenever $\mathrm{ht}\mathfrak{p}\le j$. Condition $(S_j)$ means that $\mathrm{depth}{R}_{\mathfrak{p}}\ge \min \{j,\dim R_{\mathfrak{p}}\}$ for every prime $\mathfrak{p}$. A finite module $M$ satisfies $(S_j)$ if ${\mathrm{depth}}_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}\ge \min \{j,\dim {\mathrm{Supp}}_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}\}$ for every prime in its support. Outside the support the condition is vacuous, consistent with depth of the zero module being $+\mathrm{\infty }$ and the empty support having no nonnegative dimension. Thus the zero module satisfies all $(S_j)$ conditions, and the zero ring satisfies both families vacuously.

Every submodule of a finite module over a Noetherian ring has a minimal primary decomposition: Assume Dependent Choice. Let $R$ be a Noetherian commutative ring and let $M$ be a finitely generated left $R$-module. Every submodule $N\subseteq M$ has a finite primary decomposition. After deleting redundant components and combining equal radicals, one obtains a minimal primary decomposition. When $N=M$, the decomposition is the empty intersection, interpreted as $M$. In particular, every ideal of a Noetherian ring has a minimal primary decomposition.

The radicals in a minimal primary decomposition are exactly the associated primes of the quotient: Let $R$ be a Noetherian commutative ring, let $M$ be a finitely generated left $R$-module, and let $N=Q_1\cap \cdots \cap Q_r$ be a minimal primary decomposition in which each $Q_i$ is ${\mathfrak{p}}_i$-primary. Assume each ${\mathfrak{p}}_i$ is a prime ideal. Then ${\mathrm{Ass}}_R(M\mathrm{/}N)=\{{\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_r\}\mathrm{.}$

Depth drops by one after quotienting by a regular element: Let $R$ be Noetherian, let $M$ be finite, let $I$ lie in the Jacobson radical, and let $x\in I$ be $M$-regular. Then ${\mathrm{depth}}_I(M\mathrm{/}xM)={\mathrm{depth}}_I(M)-1.$

The local depth-zero associated-prime criterion: Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M\mathrm{\ne }0$ be a finite $R$-module. Then $\mathrm{depth}(M)=0⟺\mathfrak{m}\in {\mathrm{Ass}}_R(M)\mathrm{.}$

Associated primes commute with localization for finite modules: Let $R$ be a Noetherian commutative ring, let $M$ be a finitely generated left $R$-module, and let $S\subseteq R$ be multiplicative. Then ${\mathrm{Ass}}_{S^{-1}R}(S^{-1}M)=\{S^{-1}\mathfrak{p}:\mathfrak{p}\in {\mathrm{Ass}}_R(M),\mathfrak{p}\cap S=\mathrm{\varnothing }\}\mathrm{.}$

A nonzero module over a Noetherian ring has an associated prime: Under Dependent Choice, if $R$ is a Noetherian commutative ring and $M$ a nonzero left $R$-module, then ${\mathrm{Ass}}_R(M)$ is nonempty.