Noetherian local ring of positive dimension has element in maximal ideal outside minimal primes

This fact is an application of the following pivotal fact/result/idea: prime avoidance lemma
View other applications of prime avoidance lemma OR Read a survey article on applying prime avoidance lemma

Statement

Suppose $(A,\mathfrak{m})$ is a Noetherian local ring that is not Artinian: in other words, the unique maximal ideal $\mathfrak{m}$ is not a minimal prime. Then, there exists an element $x\in \mathfrak{m}$ such that $x$ is not in any minimal prime.

For a reduced ring, i.e. a ring with no nilpotents, it is also clearly true that any element outside the union of minimal primes must be a nonzerodivisor. While this may not be true in general for arbitrary rings, the effect as far as Hilbert-Samuel polynomials goes, is the same as for a nonzerodivisor.

Facts used

• In a Noetherian ring, every prime contains a minimal prime, and there are only finitely many minimal primes
• Prime avoidance lemma