# Thread: Proving a quotient ring is a field

1. ## 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.

2. Originally Posted by flabbergastedman
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

3. oops! I will change the opening post, but yes, the problem is merely to show whether or not R/I is a field.

4. 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