Equivalence of Definitions of Unital Associative Commutative Algebra/Correspondence
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 |
$\blacksquare$