Definition:Unital Associative Commutative Algebra

Definition
Let $R$ be a commutative ring with unity.

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 $\struct {A, *}$ 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 $R$, $\struct {A, f}$, 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.

Also see

 * Equivalence of Definitions of Unital Associative Commutative Algebra
 * Definition:Unital Associative Commutative Algebra Homomorphism