# Definition:Unital Algebra

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## Definition

Let $R$ be a commutative ring.

Let $\left({A, *}\right)$ be an algebra over $R$.

Then $\left({A, *}\right)$ is a **unital algebra** if and only if the algebraic structure $\left({A, \oplus}\right)$ has an identity element.

That is:

- $\exists 1_A \in A: \forall a \in A: a * 1_A = 1_A * a = a$

## Notation

The unit of the algebra is usually denoted $1$ when there is no source of confusion with the unit of $R$ (if it is a ring with unity).

## Also known as

The term **unitary algebra** is also encountered, but this should not be confused with unitary Lie algebra and other notions related to a unitary group.

## Also defined as

Some sources use the definition of a **unital** algebra over a field as what is to be understood when the term **algebra** is used.