# Proving a quotient ring is a field

• Nov 16th 2009, 02:50 PM
flabbergastedman
Proving a quotient ring is a field
We are given R/I, where

R:= {$\displaystyle \left(\begin{array}{cc}q&r\\0&s\end{array}\right)$: q,r,s are in the rational numbers} and I:={$\displaystyle \left(\begin{array}{cc}0&r\\0&s\end{array}\right)$: 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.
• Nov 16th 2009, 03:36 PM
tonio
Quote:

Originally Posted by flabbergastedman
We are given R/I, where

R:= {$\displaystyle \left(\begin{array}{cc}q&r\\0&s\end{array}\right)$: q,r,s are in the rational numbers} and I:={$\displaystyle \left(\begin{array}{cc}0&r\\0&s\end{array}\right)$: 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 $\displaystyle R\slash I$ is a field. Well, define $\displaystyle f: R\rightarrow \mathbb{Q}$ by $\displaystyle f\left(\begin{array}{cc}q&r\\0&s\end{array}\right) =q$ , and now just check that f is a ring homom. and $\displaystyle I=Ker(f)$

Tonio
• Nov 16th 2009, 03:38 PM
flabbergastedman
oops! I will change the opening post, but yes, the problem is merely to show whether or not R/I is a field.
• Nov 16th 2009, 03:46 PM
flabbergastedman
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