Proposition

Show that $N$ can be arbitrarily large in $I:J^{\infty} = I:J^N$.

Solution

Let $N \in \mathbb{N}$ be given. Let $I = \ev{x^N(y - 1}, J = \ev{x}$. Then clearly, $I:J^{\infty} = I:J^N$.

Then $y - 1 \in I:J^{\infty}$ because $\forall a \in J, (y - 1)a^N \in I$. Let $1 \leq n \leq N - 1$. Then $y - 1 \notin I:J^{n}$ because $x^n \in J^n$ and $(y - 1)x^n \notin I$.

Therefore, $N$ is the smallest integer such that $I:J^{\infty} = I:J^N$, and $N$ was given arbitrarily, so $N$ can be arbitrarily large.