A (unital associative) ring
is a set
equipped with two binary operations
(addition (``+'') and multiplication (``
'')) such that
the following axioms are satisfied.
is an additive group with respect to the addition.
- distributive law holds. Namely, we have
- The multiplcation is associative.
has a multiplicative unit.
In this lectuer we are mainly interested in commutative rings,
that means, rings on which the multiplication satisfies the commutativity law.
For any ring
, we denote by
the zero element of
(respectively, the unit element of
are elements of
the following rules.
When no confusion arises, we omit the subscript `
from a unital associative ring
to another unital associative ring
is said to be ring homomorphism
if it satisfies the following conditions.
Our aim is to show the following.
Any regular local ring is UFD.
A commutative ring
is said to be a local ring if it has only one
be a local ring. Then the maximal ideal of
- A commutative ring
is a local ring if and only if
of non-units of
forms an ideal of