# Definition:Unital Associative Commutative Algebra

## 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:

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