# Show the image of an ideal is an ideal of the image

1. May 4, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Let $\mu : R \to R'$ be a ring homomorphism and let $N$ be an ideal of $R$. Show that $\mu [N]$ is an ideal of $\mu[R]$.

2. Relevant equations

3. The attempt at a solution
For something to be an ideal of a ring it must be an additive subgroup $N$ such that $aN \subseteq N$ and $Nb \subseteq N$ for all $a,b \in R$.

Now, I know that $\mu [N]$ is a subgroup of $R$ under addition, but I don't necessarily know that it is a subgroup of $\mu [R]$. How can I proceed if I can't establish this?

Last edited: May 4, 2017
2. May 5, 2017

### andrewkirk

If a set A is a group and it is a subset of another group B, with the same group operation, it is a subgroup of B. Let $A=\mu(N),\ B=\mu(R)$. First you need to show that $\mu(R)$ is a group. then you need to show that the criteria of the first sentence are met.