Thread: Algebra, Problems For Fun (40)

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

Hint:

Spoiler:

has only one maximal ideal.

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.

Originally Posted by TheAbstractionist

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.

contains the identity element of so if it was an ideal of it would be equal to but it's not! so is not an ideal of

you can also see that by checking that is not closed under multiplication by elements of

Originally Posted by NonCommAlg

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

Hint:

Spoiler:

has only one maximal ideal.

(I've used , with the equivalent multiplication, because it's quicker to type...)

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.