Equivalence of Definitions of Unital Associative Commutative Algebra/Correspondence

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Theorem

Let $A$ be a commutative ring with unity.

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 $f$.


Proof

Let $\cdot: A \times B \to B$ the ring action of $B$.


$(1)$ implies $(2)$

Let $C$ equal the underlying ring of $B$ and $f: A \to C$ equal the canonical mapping to the unital algebra $B$.

We show that $B$ is the algebra defined by $f$.


Addition

By definition of the underlying ring of $B$, the addition of $C$ is the addition of $B$, say $+$.

By definition of the module defined by $f$, the addition of the algebra defined by $f$ is also $+$.


Multiplication

By definition of the underlying ring of $B$, the multiplication of $C$ is the ring product of $B$, say $\times$.

By definition of the algebra defined by $f$, its multiplication is also $\times$.


Ring action

It remains to show that the ring action $\cdot$ of $B$ is the ring action $*$ of the module defined by $f$.

We have, for $a \in A$ and $b \in B$:

\(\ds a * b\) \(=\) \(\ds \map f a \times b\) Definition of Module Defined by Ring Homomorphism
\(\ds \) \(=\) \(\ds \paren {a \cdot 1_B} \times b\) Definition of Canonical Homomorphism from Ring to Unital Algebra
\(\ds \) \(=\) \(\ds a \cdot \paren {1_B \times b}\) Definition of Bilinear Mapping
\(\ds \) \(=\) \(\ds a \cdot b\) Definition of Unit of Algebra

$\Box$


2 implies 1

Let $B$ equal the algebra defined by $f$.


Addition

By definition of the module defined by $f$, the addition of $B$ is the addition of $C$, say $+$.

By definition of the underlying ring of $B$, its addition is also $+$.


Multiplication

By definition of the algebra defined by $f$, the multiplication of $B$ is the ring product of $C$, say $\times$.

By definition of the underlying ring of $B$, its multiplication is also $\times$.

Thus $C$ is the underlying ring of $B$.


Homomorphism

By Identity is Unique, the unit $1_B$ of $B$ equals the unity $1_C$ of $C$.

Let $g: A \to B$ be the canonical mapping.

We show that $g = f$.

We have, for $a \in A$:

\(\ds \map g a\) \(=\) \(\ds a \cdot 1_B\) Definition of Canonical Homomorphism from Ring to Unital Algebra
\(\ds \) \(=\) \(\ds a \cdot 1_C\) $1_B = 1_C$
\(\ds \) \(=\) \(\ds \map f a \times 1_C\) Definition of Module Defined by Ring Homomorphism
\(\ds \) \(=\) \(\ds \map f a\) Definition of Unity of Ring

$\blacksquare$

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