# Equivalence of Definitions of Unital Associative Commutative Algebra

## 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 $\struct {B, f}$ and $\struct {C, g}$ be rings under $A$.

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

The following are equivalent:

$(1): \quad h$ is a morphism of rings under $A$.
$(2): \quad h$ 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.