Let $A$ be a commutative ring with $1$.We say that $A$ is coherent if and only if every finitely generated ideal of $A$ is finitely presented.

Does there exist a coherent ring such that nil-radical of $A$ is NOT finitely generated?

In other words, by the definition of coherent rings, nil-radical of $A$ is not finitely presented!

Actually, my question is, Does there exist a coherent structure sheaf $ \mathcal O_X$ such that nilradical sheaf $ \mathcal N_X$ is not coherent?