Applications of the prime avoidance lemma

The proof of the First and Second Uniqueness Theorems of Primary Decomposition uses prime avoidance lemma in an essential way.

See Atiyah-Mac Donald, Introduction to Commutative Algebra, Chapter 4 (in that book prime avoidance lemma is referred as Proposition 1.11).

Prime avoidance can be used to show the following fundamental result on regular sequences:

If $R$ is a noetherian ring, $\mathfrak{a}\subseteq R$ is an ideal, and $M$ is an $R$-module of finite type, then every maximal $M$-sequence in $\mathfrak{a}$ has length equal to the $\mathfrak{a}$-depth of $M$.

It can also be used to show the following:

Regular local rings are integral.

Theorem. For a given commutative ring $R$, then $Min(R)$, the set of minimal prime ideals of $R$, is a finite set if and only if no minimal prime ideal of $R$ is contained in the union of the remaining minimal primes.

Sketch of Proof. The implication $\Rightarrow$ of the above nice result is deduced from the prime avoidance lemma. The reverse implication is deduced from the fact that $Min(R)$ is quasi-compact with respect to the flat topology.

Remember that the collection of $V(f)=\{\mathfrak{p}\in Spec(R):f\in\mathfrak{p}\}$ with $f\in R$ forms a sub-base for the opens of the flat topology over $Spec(R)$.

For more details on the flat topology see arXiv:1609.00947 or arXiv:1503.04299.