I have been thinking about this problem.

If I get bored I might start working on it but the idea is this.

First, R is a ring. That is simple to show just show that all the ring defintions are satisfied. Lengthy, but easy.

Second, to show that I is an ideal of R is also not no hard. Show that I is an additive subgroup. And then show that aI and Ia are subsets of I. However, if you are smart you can skip this step and go to step 3 which will answer this question.

Third, is the tricky part. To show the Isomorphism between R/I and F x F x F. What you need is to define a ring homomorphism from R to F x F x F in such a way that the kernel is I and that phi [R]=F x F x F. Then by the fundamental homomorphism theorem we have not only shown that I is an ideal but settled that R/I is isomorphic with F x F x F.