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Commutative algebra

Yoshifumi Tsuchimoto

$% latex2html id marker 573 \fbox{01.Review of elementary definitions on modules.}$

DEFINITION 01.1 A (unital associative) ring is a set

equipped with two binary operations (addition (``+'') and multiplication (`` $\cdot$ '')) such that the following axioms are satisfied.

Ring-1.

is an additive group with respect to the addition.

Ring-2.

distributive law holds. Namely, we have

$% latex2html id marker 626 $\displaystyle a(b+c)=ab + bc,\quad (a+b)c=ac+bc \qquad (\forall a,\forall b,\forall c\in R). $$

Ring-3.

The multiplcation is associative.

Ring-4.

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 (respectively, ) the zero element of (respectively, the unit element of ). Namely, and are elements of characterized by the following rules.

, $\forall a \in R$ .
$a \cdot 1_R =a$ , $1_R \cdot a =a$ $\forall a \in R$ .

When no confusion arises, we omit the subscript ` ${}_R$ ' and write

instead of

.

DEFINITION 01.2 A map $R\to S$ from a unital associative ring

to another unital associative ring

is said to be ring homomorphism if it satisfies the following conditions.

Ringhom-1.
Ringhom-2.
Ringhom-3.

Our aim is to show the following.

THEOREM 01.3 Any regular local ring is UFD.

DEFINITION 01.4 A commutative ring

is said to be a local ring if it has only one maximal ideal.

EXAMPLE 01.5 We give examples of local rings here.

Any field is a local ring.
For any commutative ring and for any prime ideal $\mathfrak{p}\in \operatorname{Spec}(A)$ , the localization $A_\mathfrak{p}$ is a local ring with the maximal ideal $\mathfrak{p}A_\mathfrak{p}$ .

LEMMA 01.6

Let be a local ring. Then the maximal ideal of coincides with $A\setminus A^\times$ .
A commutative ring is a local ring if and only if the set $A\setminus A^\times$ of non-units of forms an ideal of .

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2012-04-13