I was wondering the following:
Let $F$ be a field, consider the polynomial ring $R = F[x_1, \ldots, x_n]$, then it is a vector space over $F$. Given an ideal $I$ of $R$, what are sufficient and necessary conditions on the ideal $I$ for $R/I$ to have finite dimension over $F$?
For example: $F[x,y]/<x^2,y^2>$ has finite dimension, but $F[x,y]/<xy>$ does not (?)
Please do correct me if the question doesn't make sense at all.