Associated prime ideals and regular elements in polynomial rings

This post is here to record a fact that ought to be better known and for which I ought to have known a proper proof beforehand. Let's start by defining two thirds of the terms of the title. By polynomial ring , I simply mean a ring of polynomials $A[T_1,T_2,\dots]$ over a commutative ring $A$, in any number of indeterminates. The most important case here will however be that of a ring of…