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# Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 01.1   A (unital associative) ring is a set equipped with two binary operations (addition (+'') and multiplication ( '')) 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

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.

• ,      .
• ,      .
When no confusion arises, we omit the subscript ` ' and write instead of .

DEFINITION 01.2   A map 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 , the localization is a local ring with the maximal ideal .

LEMMA 01.6
1. Let be a local ring. Then the maximal ideal of coincides with .
2. A commutative ring is a local ring if and only if the set of non-units of forms an ideal of .

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2012-04-13