COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

01.Review of elementary definitions on modules.

Definition 1.1. A (unital associative) ring is a set R equipped with two binary operations (addition (“+”) and multiplication (“·”)) such that the following axioms are satisfied.

(Ring-1) R is an additive group with respect to the addition. (Ring-2) distributive law holds. Namely, we have

a(b + c) = ab + bc, (a + b)c = ac + bc (∀a, ∀b, ∀c ∈ R).

(Ring-3) The multiplcation is associative. (Ring-4) 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.

• a + 0_{R} = a, 0_{R} + a = a ∀a ∈ R.

• a · 1_{R} = a, 1_{R} · a = a ∀a ∈ R.

When no confusion arises, we omit the subscript ‘_{R}’ and write 0, 1 instead of 0_{R}, 1_{R}.

Definition 1.2. A map R → 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.

(Ringhom-1) f(a + b) = f(a) + f(b) (Ringhom-2) f(ab) = f(a)f(b) (Ringhom-3) f(1_{R}) = 1_{S}

Our aim is to show the following.

Theorem 1.3. Any regular local ring is UFD.

Definition 1.4. A commutative ring A is said to be a local ring if it has only one maximal ideal.

Example 1.5. We give examples of local rings here.

• Any field is a local ring.

• For any commutative ring A and for any prime ideal 𝔭 ∈ Spec(A), the localization A_{𝔭} is a local ring with the maximal ideal 𝔭A_{𝔭}.

Lemma 1.6. (1) Let A be a local ring. Then the maximal ideal

of A coincides with A \ A^{×}.

(2) A commutative ring A is a local ring if and only if the set A\A^{×} of non-units of A forms an ideal of A.