Problem: Let monomial ideal $I = \langle x^3,y^3 \rangle \subseteq K[x,y]$, consider quotient ring $V = K[x,y]/I$ as a $K$-vector space. Calculate the dimension of $K$-vector space $V$.
Could you give me some hint to solve this problem. Thank all!
2026-03-25 09:26:38.1774430798
Calculate the dimension of $K$-vector space $V$
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Well, the dimension is the number of standard monomials in the quotient ring, here $1,x,y,x^2,xy,y^2,x^2y,xy^2,x^2y^2$ taken modulo $I$.