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 $\struct {A, *}$ 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 $\struct {A, f}$ 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:

The algebra is viewed as a ring.

For the detailed statements, see Equivalence of Definitions of Unital Associative Commutative Algebra.


Also see

  • Results about unital associative commutative algebras can be found here.