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Math Help - Proving a quotient ring is a field

  1. #1
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    Proving a quotient ring is a field

    We are given R/I, where

    R:= { \left(\begin{array}{cc}q&r\\0&s\end{array}\right) : q,r,s are in the rational numbers} and I:={ \left(\begin{array}{cc}0&r\\0&s\end{array}\right): r,s also in the rational numbers}

    and the defined set of I forms an ideal of R
    Prove (or disprove) that R/I is a field.

    I'm not sure where to even start. I know for other quotient rings, it was a question of finding factors of the "denominator" - if any existed, then it could not be a field since it would not be irreducible. But since we're dealing with an ideal, I'm not so sure. I've tried finding zero-divisors, but at this point it seems a fruitless exercise, so I turn to your collective expertise.
    Last edited by flabbergastedman; November 16th 2009 at 04:39 PM. Reason: b,c = r, s in my world.
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  2. #2
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    Quote Originally Posted by flabbergastedman View Post
    We are given R/I, where

    R:= { \left(\begin{array}{cc}q&r\\0&s\end{array}\right) : q,r,s are in the rational numbers} and I:={ \left(\begin{array}{cc}0&r\\0&s\end{array}\right): r,s also in the rational numbers}

    and the defined set of I forms an ideal of R


    I'm not sure where to even start. I know for other quotient rings, it was a question of finding factors of the "denominator" - if any existed, then it could not be a field since it would not be irreducible. But since we're dealing with an ideal, I'm not so sure. I've tried finding zero-divisors, but at this point it seems a fruitless exercise, so I turn to your collective expertise.

    You can check: you didn't ask anything, but I guess that you need to show that the quotient ring R\slash I is a field. Well, define f: R\rightarrow \mathbb{Q} by f\left(\begin{array}{cc}q&r\\0&s\end{array}\right)  =q , and now just check that f is a ring homom. and I=Ker(f)

    Tonio
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  3. #3
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    oops! I will change the opening post, but yes, the problem is merely to show whether or not R/I is a field.
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  4. #4
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    Also, I'm a little unclear as to how that proves we have a field*. if we have a mapping of f: R -> S, isn't that simply showing that the quotient ring is isomorphic to the mapping?

    edited for confusing Rings and Fields again
    Last edited by flabbergastedman; November 16th 2009 at 06:04 PM.
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