Equivalence of Definitions of Unital Associative Commutative Algebra

Theorem
Let $A$ be a commutative ring with unity.

Correspondence
Let $(B, +)$ be an abelian group.

Let $(B, +, \cdot, *)$ be a algebra over $A$ that is unital, associative and commutative.

Let $(B, +, *, f)$ be a ring under $A$.


 * 1) The underlying module $(B, +, \cdot)$ is the module structure as a ring under $A$ via $f$.
 * 2) The ring homomorphism $f : A \to B$ is the canonical mapping to the unital algebra $B$.