TITLE: Is the number of isomorphism classes of quotients of a finite dimensional commutative ring over a field finite?
QUESTION [5 upvotes]: If $A$ is a finite dimensional unital and commutative algebra over some infinite field $k$, what is the number of isomophism classes of rings of the form $A/I$ where $I$ is a proper ideal of $A$? Is it finite?
Certainly, for dimensions 7 and below (by the results of this paper) the answer to the finite question will be yes, as the number of isomorphism classes of algebras of dimension $n \leq 6$ is finite. Is the number of isomorphism classes of such rings still finite if the dimension of $A$ is greater than 7?

REPLY [2 votes]: The infinitely many $7$-dimensional algebras described in Poonen's paper are all quotients of the finite dimensional algebra $A=k[w,x,y,z]/\mathfrak{m}^3$ (where $\mathfrak{m}$ is the ideal $(w,x,y,z)$). So $A$ is a $15$-dimensional counterexample.