$I/N$ is finitely presented module

Question

Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.

My question is: Suppose that $I/N$ is a finitely presented ideal of $R/N$. Does it follow that $I$ is a finitely presented ideal of $R$?

Thanks in advance for your answer.

Answer 1

No. A counter example is the ring k[x_1, x_2, ...]/(x_ix_j) with I = N.