A local ring
is a commutative
ring containing a unique maximal ideal
. That is,
every ideal of a ring R
(except for R
itself) is contained
within a single ideal I
The most common examples of local rings are constructed from the integers:
- Z / p^nZ, where p is a prime, is a local ring;
the ideal generated by p is the maximal ideal
- the integers localized at the ideal generated by any prime p
(for example, 3Z) is a local ring. This ring is just the ring of
all fractions whose denominators are divisible by p.
One interesting property of local rings is that every element not in the unique
maximal ideal has an inverse (is invertible), while every element in the ideal does
not have an inverse.
The idea behind studying local rings is that by studying local information, you can
get information about the global ring. Many problems in algebraic number theory about
rings in general can be solved by reducing the problem to the case of local rings.