PROOF..
(1) Assume

is a local ring with the maximal ideal

.
Then for any element

,
an ideal

is an ideal of

.
By Zorn's lemma, we know that

is contained in a maximal ideal of

.
From the assumption, the maximal ideal should be

.
Therefore, we have
which shows that
The converse inclusion being obvious (why?), we have
(2) The “only if” part is an easy corollary of (1).
The “if” part is also easy.
DEFINITION 07.49
Let

be local rings
with maximal ideals

respectively.
A local homomorphism

is a homomorphism which
preserves maximal ideals. That means, a homomorphism

is said to be local
if