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# Proving R/I is a field iff I is maximal. watch

1. I can prove it one way but I can't seem to prove the converse, here is what I have so far any suggestions for the converse?

Theorem:

Let be a commutative ring with one and let be an ideal of then is a field if and only if is maximal.

Proof:

Let and so that is a non-zero element in we want to show there exists an element such that (the one in ).

Now let and note is in fact an ideal and (a proper subset since but ) and so since is maximal.

Hence the one in is also in so and for some and so which is the one in so indeed is a field.

Stuck on how to prove the converse: If is a field then is a maximal ideal of .
2. (Original post by poorform)
Stuck on how to prove the converse: If is a field then is a maximal ideal of .
Okay, so we have that is a field and suppose is an ideal of that properly contains . Suppose also that but . What can you say about ?

Once you've done that, you should be able to state that there exists some element such that .

Does this help at all? More in the spoiler:
Spoiler:
Show
Since then , so now do some stuff, can you see what? To show that .
3. got it now thanks for help!

the only step I couldn't see was to say 1-bc is in I (so in B) and bc in B (property of ideals) then ((1-bc)+bc) is in B since it is an ideal and closed under addition hence the one of the ring is in B and then it almost trivial to show B is a subset of R and R is a subset of B so B=R and so I is maximal.
4. (Original post by poorform)
got it now thanks for help!
Great. No problem!
5. (Original post by Zacken)
Great. No problem!
A very nice proof .

It won't let me rep you tho.
6. (Original post by poorform)
the only step I couldn't see was to say 1-bc is in I (so in B) and bc in B (property of ideals) then ((1-bc)+bc) is in B since it is an ideal and closed under addition hence the one of the ring is in B and then it almost trivial to show B is a subset of R and R is a subset of B so B=R and so I is maximal.
Ah, so you were basically pretty much all there except for one single step. Very nice proof indeed. :-)
7. (Original post by Zacken)
Okay, so we have that is a field and suppose is an ideal of that properly contains . Suppose also that but . What can you say about ?

Once you've done that, you should be able to state that there exists some element such that .

Does this help at all? More in the spoiler:
Spoiler:
Show
Since then , so now do some stuff, can you see what? To show that .
When you said that you had an offer for maths at Cambridge, did you mean a job offer?
8. (Original post by atsruser)
When you said that you had an offer for maths at Cambridge, did you mean a job offer?
He will eventually replace DFranklin ....
9. (Original post by TeeEm)
He will eventually replace DFranklin ....
Speaking of him, I haven't seen him around for ages. I hope he's alright.

Posted from TSR Mobile
10. (Original post by Krollo)
Speaking of him, I haven't seen him around for ages. I hope he's alright.

Posted from TSR Mobile
DFranklin
gone

ghostwalker
gone

Davros
gone

Firegarden
gone

smaug123
gone

TenOfThem
gone

Noble
gone

James........some number which I do not remember
gone

Maybe there is a "Bermuda triangle" in here somewhere, and I begin to worry if I will be gone soon ...
11. (Original post by TeeEm)
DFranklin
gone

ghostwalker
gone

Davros
gone

Firegarden
gone

smaug123
gone

TenOfThem
gone

Noble
gone

James........some number which I do not remember
gone

Maybe there is a "Bermuda triangle" in here somewhere, and I begin to worry if I will be gone soon ...
Even after you are gone from this site you will live on.

Many many years from now people will be using your legendary materials to practise Stokes, flux integrals, divergence theorem, Fourier series, PDE's etc.
12. (Original post by atsruser)
When you said that you had an offer for maths at Cambridge, did you mean a job offer?
zacken just kills it tbf. how long until he leaves this site for mathstack then stackoverflow.
13. TeeEm The new will replace the old. It is a fact of life.

(Original post by poorform)
zacken just kills it tbf. how long until he leaves this site for mathstack then stackoverflow.
And Quora
14. (Original post by Kvothe the arcane)
The new will replace the old. It is a fact of life.
Oldie power! We will resist!
15. (Original post by poorform)
zacken just kills it tbf. how long until he leaves this site for mathstack then stackoverflow.
To be fair: http://math.stackexchange.com/users/161779/zain-patel
16. A cleaner method is by contrapositive:

Let be the natural projection. If is not a field, let non-zero and non-invertible. Then is a proper non-trivial ideal of so is a proper ideal of which properly contains so is not maximal.

Conversely, if then is a non-trivial proper ideal of so is not a field.

Hey what is my integral doing there
17. (Original post by Lord of the Flies)
A cleaner method is by contrapositive: [...]
That is gorgeous.

Hey what is my integral doing there
t'was a really cool one, was looking for more approaches. Hope you don't mind.
18. (Original post by Zacken)
Hope you don't mind.
Not at all!

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