I have a theorem which says:
If the idealis maximal, then
is a field.
Is the converse true, can you say that ifis a field, then
is maximal??
Thanks!
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adie_raz
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- 26-03-2011 11:54
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- 26-03-2011 13:46
(Original post by adie_raz)
I have a theorem which says:
If the idealis maximal, then
is a field.
Is the converse true, can you say that ifis a field, then
is maximal??
Thanks!
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 thatand 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). -
adie_raz
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- 26-03-2011 14:34
(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 thatand 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).
Thank you
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