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Commutative algebra

Yoshifumi Tsuchimoto

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\fbox{01.Review of elementary definitions on modules.}

DEFINITION 01.1   A (unital associative) ring is a set $ R$ equipped with two binary operations (addition (``+'') and multiplication (``$ \cdot$ '')) such that the following axioms are satisfied.
$ R$ is an additive group with respect to the addition.
distributive law holds. Namely, we have

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

The multiplcation is associative.
$ R$ 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 $ R$ , we denote by $ 0_R$ (respectively, $ 1_R$ ) the zero element of $ R$ (respectively, the unit element of $ R$ ). Namely, $ 0_R$ and $ 1_R$ are elements of $ R$ characterized by the following rules.

When no confusion arises, we omit the subscript `$ {}_R$ ' and write $ 0,1$ instead of $ 0_R,1_R$ .

DEFINITION 01.2   A map $ R\to S$ from a unital associative ring $ R$ to another unital associative ring $ S$ is said to be ring homomorphism if it satisfies the following conditions.
$ f(a+b)=f(a)+f(b)$
$ f(ab)=f(a) f(b)$
$ f(1_R)=1_S$

Our aim is to show the following.

THEOREM 01.3   Any regular local ring is UFD.

DEFINITION 01.4   A commutative ring $ A$ is said to be a local ring if it has only one maximal ideal.

EXAMPLE 01.5   We give examples of local rings here.

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

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