This is the last problem of this series of problems that I've been posting for a while now:
Let be a field and It's easy to see that is a commutative ring with identity element. Find all maximal ideals of
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.