regular local rings are domains and cohen macaulay

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

regular local domain induction: Every regular local ring is an integral domain.

regular local parameter is nonzerodivisor: In a positive-dimensional regular local ring, every member of a regular system of parameters is a nonzerodivisor.

regular local quotient by parameter is regular: Let $(R,\mathfrak{m},k)$ be regular local of dimension $d$, and let $x\in \mathfrak{m}\setminus {\mathfrak{m}}^2$. Then $R\mathrm{/}(x)$ is regular local, of dimension and embedding dimension $d-1$.

Cohen--Macaulay local modules and rings: Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $M$ be a nonzero finite $R$-module. The module $M$ is Cohen--Macaulay when $\mathrm{depth}(M)=\dim {\mathrm{Supp}}_R(M)\mathrm{.}$ The zero module is excluded from this local definition. The local ring $R$ is Cohen--Macaulay when it is Cohen--Macaulay as an $R$-module.

Regular Sequence On A Module: Let $R$ be a commutative unital ring, let $M$ be an $R$-module, and let $\mathbf{x}=(x_1,\dots ,x_n)$ be a finite ordered sequence in $R$. The sequence is $M$-regular when $M\mathrm{/}(x_1,\dots ,x_{i-1})M\mathrm{\ne }0$ and multiplication by $x_i$ is injective on it for every $i$, and $M\mathrm{/}(\mathbf{x})M\mathrm{\ne }0$.

Depth is bounded by support dimension: For every nonzero finite module $M$ over a Noetherian local ring $R$, $0\le {\mathrm{depth}}_R(M)\le \dim {\mathrm{Supp}}_R(M)\mathrm{.}$ The nonzero hypothesis is essential for this formulation: under the adopted convention ${\mathrm{depth}}_R(0)=+\mathrm{\infty }$, whereas the empty support has no nonnegative Krull dimension.

Depth with respect to an ideal: For a finite module $M$ with $IM\mathrm{\ne }M$, ${\mathrm{depth}}_I(M)$ is the supremum of the lengths of $M$-regular sequences in $I$; for a local ring depth means depth with respect to its maximal ideal.