Ideal of ideal needs not to be an ideal Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with coefficient from a field, thanks

Take $R := \mathbb Q[x]$, $I := x\, \mathbb Q[x]$ and $J := \{a_1 x + a_2 x^2 + \cdots + a_n x^n : a_1\in\mathbb Z\}$. Then $I$ is an ideal of $R$ and $J$ is an ideal of $I$. But $J$ is not an ideal of $R$.
