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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.