Definition:Unital Algebra

Definition
Let $R$ be a commutative ring.

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

Then $\left({A_R, \oplus}\right)$ is a unitary algebra the algebraic structure $\left({A_R, \oplus}\right)$ has an identity element.

That is:
 * $\exists 1_A \in A_R: \forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$

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 unital algebra is also encountered.

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