Clearly is an ideal of and the quotient ring is a commutative ring with identity
Let’s let denote the element in
Then if Hence is a field and so is a maximal ideal.
The subring with is clearly an ideal: .
However, this ideal is also maximal. This is because if is an ideal containing an element , , then and so .
Subsequently this ideal contains all other proper ideals, as if it didn't then there would be an ideal with , a contradiction. Thus it is the only maximal ideal.