Last update: 29 March 2014
Let be a commutative ring.
- A prime ideal is an ideal of such that is an integral domain.
- A maximal ideal is an ideal of such that is a field.
- A local ring is a ring with a unique maximal ideal.
- A multiplicative subset is a subset which is closed under multiplication.
Exercise: An ideal of is prime if and only if is a multiplicative subset of .
Let be a commutative ring, let be a subset of and let be a prime ideal of .
- The multiplicative closure of is the smallest multiplicative subset such that .
- The ring of fractions defined by is the ring
with operations given by
and with the map
- The localization of at is
Let be a ring, let be a prime ideal of and let
be the localization of at .
- The functor
- The map
is a bijection which takes prime ideals to prime ideals and
is the unique maximal ideal of
Let be a prime ideal of contained in . Then is an integral domain and
is an integral domain???
This proof needs to be filled in.
Notes and References
Where are these from?