# What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version.

Ill-Defined Question (IDQ). Let $$R$$ be a (commutative, unital) Noetherian ring, $$M$$ a finitely generated $$R$$-module. What is the probability that a randomly chosen element $$m\in M$$ has prime annihilator (i.e. $$\operatorname{ann}_R(m)$$ is a prime ideal)?

This is ill-defined because there's no obvious way to assign meaning to "random" and "probability" in this context. So, I'm going to restrict to the (very) special case that $$R$$ is a finitely-generated $$\Bbb Z$$-algebra, and for simplicity, I'm going to assume that $$M=R$$.

Well-Defined Question (WDQ). Let $$R$$ be a finitely-generated (commutative, unital) $$\Bbb Z$$-algebra. Fix a generating set $$x_1,\ldots,x_n$$, and let $$R_k:= \{y \in R : y=f(x_1,\ldots,x_n)\text{ for some polynomial }f \text{ of degree }\leq k\}.$$ Set $$P_k := \frac{|\{y\in R_k : \operatorname{ann}_R(y)\text{ is a prime ideal}\}|}{|R_k|}.$$ What can be said about $$\lim_{k\to\infty} P_k$$? Is it positive? Does it depend on the chosen generating set?