Let R be a commutative ring and I an ideal of R. Show that if I is maximal then R/I is a field. I'm a bit stuck on how to start this. Any would would be appreciated. Thanks.
Let R be a commutative ring and I an ideal of R. Show that if I is maximal then R/I is a field. I'm a bit stuck on how to start this. Any would would be appreciated. Thanks.
Suppose R/I is not a field. Then there exists x+I in R/I with no multiplicative inverse, so...
If you know the correspondence theorem this is immediate (i.e. if R/I has a non-trivial proper ideal J then consider the corresponding ideal J' in R. The ideal J' contains I so must be either R or I by maximality of I. The first contradicts the fact that J was proper, the second contradicts non-triviality)
I am assuming therefore, that you don't know and/or aren't expected to know the correspondence theorem. In that case; it is a bit harder to think up.
Hint: For each non-zero element x+I in R/I and consider the ideal J = I + Rx.