local rings

DEFINITION 02.10   A commutative ring $A$ is said to be a local ring if it has only one maximal ideal.

EXAMPLE 02.11   We give examples of local rings here.

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

DEFINITION 02.12   Let $A,B$ be local rings with maximal ideals $\mathfrak{m}_A, \mathfrak{m}_B$ respectively. A local homomorphism $\varphi:A \to B$ is a homomorphism which preserves maximal ideals. That means, a homomorphism $\varphi$ is said to be loc al if

$$\varphi^{-1}(\mathfrak{m}_B) =\mathfrak{m}_A$$

EXAMPLE 02.13 (of NOT being a local homomorphism)  

$$\mathbb{Z}_{(p)}\to \mathbb{Q}$$

is not a local homomorphism.

&dotfill#dotfill;

LEMMA 02.14 (Zorn's lemma)   Let $\mathcal S$ be a partially ordered set. Assume that every totally ordered subset of $\mathcal S$ has an upper bound in $\mathcal S$ . Then $\mathcal S$ has at least one maximal element.