Definition:Unital Associative Commutative Algebra

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

Equivalence of Definitions
While, strictly speaking, the above definitions of a unitary algebra over a ring with unity do define different objects, they can be shown to be equivalent in a strong sense. See Equivalence of Definitions of Unitary Algebra.

Also see

 * Equivalence of Definitions of Unitary Algebra