Definition:Unital Associative Commutative Algebra
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Definition
Let $R$ be a commutative ring with unity.
Definition 1
A unital associative commutative algebra over $R$ is an algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary.
Definition 2
A unital associative commutative algebra over $R$ is a ring under $A$, that is, an ordered pair $(A, f)$ where:
- $A$ is a commutative ring with unity
- $f : R \to A$ is a unital ring homomorphism
Also known as
A unital associative commutative algebra over $R$ is also known as an $R$-algebra, as is a general algebra over $R$.
Equivalence of Definitions
While, strictly speaking, the above definitions do define different objects, they are equivalent in the following sense:
- An algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary, is identified with the ring under $R$ equal to its underlying ring together with its canonical mapping $R \to A$. The algebra is viewed as a ring.
- A ring under $R$, $(A, f)$, is identified with the algebra defined by $f$. The algebra is viewed as an algebra.
For the detailed statements, see Equivalence of Definitions of Unital Associative Commutative Algebra.