# 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.