# Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 03.1   Let be a ring. A ring is called an -algebra if there is given a distinguished ring homomorphism (which is called the structure morphism) from to .

DEFINITION 03.2   Let be a commutative ring. Let be an -algebra. Let be an -module. A map is called an -derivation if it satisfies the following conditions.
1. is -linear.
2. .

PROPOSITION 03.3   For any algebra over a ring , There exists an universal derivation .

DEFINITION 03.4   The module is called the module of differentials of over .

DEFINITION 03.5   An algebra over a ring is called unramified over if . is called étale over if it is unramified and flat.

LEMMA 03.6   Let be a commutative ring. Let be its multiplicative subset. Then is étale over .