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?


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