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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: $F[x,y]/$ has finite dimension, but $F[x,y]/$ does not (?)

Please do correct me if the question doesn't make sense at all.

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    What sort of conditions are you looking for? – 2017-02-16
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    I was just looking for an answer as general as possible. But primarily interested when $I$ is a finitely generated ideal - the conditions on elements that generate the ideal $I$. – 2017-02-16
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    I don't remember all the details, but I'm pretty sure you can compute the dimension of a quotient ring using a Gröbner basis of the ideal. – 2017-02-18
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    Iff $R/I$ is Artinian. (Look for zero-dimensional ideals.) – 2017-02-19

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