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# Maximal Ideals watch

1. I have a theorem which says:

If the ideal is maximal, then is a field.

Is the converse true, can you say that if is a field, then is maximal??

Thanks!
I have a theorem which says:

If the ideal is maximal, then is a field.

Is the converse true, can you say that if is a field, then is maximal??

Thanks!
You need R to be commutative as well don't you?

The converse is true. Perhaps you have a theorem that says:

If R is commutative with an identity element then R is a field if and only if the only ideals of R are {0} and R

Also, you need the third isomorphism theorem:

If R is a ring and I and ideal of R, there is a one to one correspondence between ideals J so that and ideals of R/I.

Then R/I is a field if and only if the ideals of R/I are R/I and {0} if and only if there are no ideals strictly between I and R (which is what it means for I to be maximal).
3. (Original post by SsEe)
You need R to be commutative as well don't you?

The converse is true. Perhaps you have a theorem that says:

If R is commutative with an identity element then R is a field if and only if the only ideals of R are {0} and R

Also, you need the third isomorphism theorem:

If R is a ring and I and ideal of R, there is a one to one correspondence between ideals J so that and ideals of R/I.

Then R/I is a field if and only if the ideals of R/I are R/I and {0} if and only if there are no ideals strictly between I and R (which is what it means for I to be maximal).
Thanks for the reply! yes I am working with commutative rings (sorry I didn't mention that).

Thank you

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Updated: March 26, 2011
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