1. Prove that

is a subring of

and that

is a maximal ideal in

.

Showing it's a subring is obvious. Just show closure under addition and multiplication and that

. Also show that the solution of the equation

is in

.

To show that

is maximal consider the following:

means that there is some

such that

where

. So

. And any ideal that contains

must contain

so that

.

Is this correct?

No! Why would J = R if J contains r + si = j??? J = R iff 1 is in J, and this is what you must show.

2. To show that

is a field we know this from the fact that

is maximal. How do you show it has 9 elements?