I have a question related to this post.
Suppose $I$ and $J$ are ideals in $R=k[x_1,\ldots,x_n]$ with $In(I)\cong In(J)$ where $In(I)$ is the ideal generated by the leading term of all those $f\in I$, with the additional assumption that the minimal number of generators generating $I$ equals the minimal number of generators generating $J$. Then doesn't that imply that $R/I$ is isomorphic to $R/J$?
Can someone give me a counter-example to this or refer me to a reference?
Thanks in advance.
Addendum: suppose $k$ is algebraically closed.