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 $\struct {C, f}$ be a ring under $A$.
The following statements are equivalent:
- $(1): \quad C$ is the underlying ring of $B$ and $f: A \to C$ is the canonical homomorphism to the unital algebra $B$.
- $(2): \quad B$ 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 statements 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.
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