**Yoshifumi Tsuchimoto**

- 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.

- , .
- , .

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

Our aim is to show the following.

- 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 .

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