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 $\mathbb{Q}$.)

Title	extended ideal
Canonical name	ExtendedIdeal
Date of creation	2013-03-22 12:55:34
Last modified on	2013-03-22 12:55:34
Owner	drini (3)
Last modified by	drini (3)
Numerical id	6
Author	drini (3)
Entry type	Definition
Classification	msc 13A15
Classification	msc 14K99
Related topic	ContractedIdeal