If is a commutative ring with identity, and is a maximal ideal, then is a field. What are the ideals of a field?

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- September 7th 2010, 06:29 AM #1

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- September 7th 2010, 07:42 AM #2

- September 7th 2010, 08:14 AM #3

- September 7th 2010, 11:56 AM #4

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that was what i had set out to prove....that if If R is a commutative ring with identity, and M is a maximal ideal, then R/M is a field.....they directly said that R/M has 2 ideals 0 and R/M itself....so i cld'nt get how they said this...

could you please tell me somethng abt this correspondence theorem. cz i dont seem to get what it says....

- September 7th 2010, 11:32 PM #5
The correspondence theorem (I believe) basically says two things,

1. Let with . Then .

2. If then there exists an ideal such that and .

So, essentially, the ideals in correspond precisely to the ideals such that . As there are no such proper ideals, as is maximal, then what can you conclude?