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Math Help - Quotient Rings and Fields

  1. #1
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    Question 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!!
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  2. #2
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    Quote Originally Posted by Kirsty View Post
    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
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