Definition:Unital Associative Commutative Algebra

Definition
Let $R$ be a commutative ring with unity.

Equivalence of Definitions
While, strictly speaking, the above definitions do define different objects, they can be shown to be equivalent in a strong sense. See Equivalence of Definitions of Unital Associative Commutative Algebra.

Also see

 * Equivalence of Definitions of Unital Associative Commutative Algebra