# extended ideal

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

It is not true in general that if $\mathfrak{a}$ is an ideal in $A$, the image of $\mathfrak{a}$ 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 $\mathbb{Q}$.)

Title extended ideal ExtendedIdeal 2013-03-22 12:55:34 2013-03-22 12:55:34 drini (3) drini (3) 6 drini (3) Definition msc 13A15 msc 14K99 ContractedIdeal