# Definition:Unital Algebra

(Redirected from Definition:Unitary Algebra)

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The validity of the material on this page is questionable.In particular: This is based on sources who only deal with unitary modules. The question is whether this should be part of the definition of a unital algebra.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Questionable}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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.