Equivalence of Definitions of Unital Associative Commutative Algebra

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Theorem

Let $A$ be a commutative ring with unity.

Correspondence

Let $B$ be a algebra over $A$ that is unital, associative and commutative.

Let $(C, f)$ be a ring under $A$.

The following are equivalent:

1. $C$ is the underlying ring of $B$ and $f : A \to C$ is the canonical mapping to the unital algebra $B$.
2. $B$ is the algebra defined by $f$.

Homomorphisms

Let $(B, f)$ and $(C, g)$ be rings under $A$.

Let $h : B \to C$ be a mapping.

The following are equivalent:

1. $h$ is a morphism of rings under $A$.
2. $h$ is a unital algebra homomorphism from the algebra defined by $f$ to the algebra defined by $g$.

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:

Object functor:		sends an algebra $(B, +, \cdot, *)$ to the ring under $A$ equal to $(B, +, *, f)$, where $f : A \to B$ is the canonical mapping to the unital algebra $B$.
Morphism functor:		identity mapping

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

Object functor:		sends an ring under $A$, $(B, +, *, f)$ to the ring under $A$ equal to the algebra $(B, +, \cdot, *)$, with underlying module $(B, +, \cdot)$ the module structure defined by $f$.
Morphism functor:		identity mapping

are inverse functors.

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