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 $B$.

Isomorphism of categories
Let $A-\operatorname{Alg}$ be the full subcategory of the category of unital algebras over $A$ consisting of commutative associative algebras.

Let $A/\mathbf{Ring}$ be the coslice category of rings under $A$.

Then the covariant functors:

$F : A-\operatorname{Alg} \to A/\mathbf{Ring}$ with:

$G : A/\mathbf{Ring} \to A-\operatorname{Alg}$ with:

are inverse functors.

In particular, $A-\operatorname{Alg}$ and $A/\mathbf{Ring}$ are isomorphic.