extended ideal

Let $f:A\to B$ be a ring map. We can look at the ideal generated by the image of $𝔞$, which is called an extended ideal and is denoted by ${𝔞}^e$.

It is not true in general that if $𝔞$ is an ideal in $A$, the image of $𝔞$ under $f$ will be an ideal in $B$. (For example, consider the embedding $f:\mathbb{Z}\to \mathbb{Q}$. The image of the ideal $(2)\subset \mathbb{Z}$ is not an ideal in $\mathbb{Q}$, since the only ideals in $\mathbb{Q}$ are $\{0\}$ and all of $\mathbb{Q}$.)