# Thread: Quotient Rings and Fields

1. ## Quotient Rings and Fields

Hi

I am struggling with the following question I have been set:

Let R be a commutative ring that contains a field F. Let I be an ideal of R with I not equal to R. Show that the quotient ring R/I contains a subring that is isomorphic to F.

I think it has something to do with isomorphism theorems (possibly 1st) but Im really not sure.

Any help would be greatly appreciated!!

2. Originally Posted by Kirsty
Hi

I am struggling with the following question I have been set:

Let R be a commutative ring that contains a field F. Let I be an ideal of R with I not equal to R. Show that the quotient ring R/I contains a subring that is isomorphic to F.

I think it has something to do with isomorphism theorems (possibly 1st) but Im really not sure.

Any help would be greatly appreciated!!

Hints:

(1) If $\mathbb{F}$ is a field and $R$ is a commutative ring (with unity, to make things simpler), then any ring homomorphism $\mathbb{F}\rightarrow R$ is either trivial or 1-1

(2) Define $f: \mathbb{F}\rightarrow R\slash I\,\,\,by\,\,\,f(k):= k+I\,,\,\,\forall\,k\in\mathbb{F}$ . If f is trivial then $\mathbb{F}\leq I\Longrightarrow\,I=R$ (why??) , so it must be that $f$ is an injection...

Tonio