Thread: Proving a quotient ring is a field

We are given R/I, where

R:= { : q,r,s are in the rational numbers} and I:={ : 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.

Originally Posted by flabbergastedman

We are given R/I, where

R:= { : q,r,s are in the rational numbers} and I:={ : 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 is a field. Well, define by , and now just check that f is a ring homom. and

Tonio

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

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