Definition:Unital Algebra

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

Then $\left({A_R, \oplus}\right)$ is a unitary algebra if it has an identity element $1_A$ called a unit for $\oplus$:
 * $\exists 1_A \in A_R: \forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$

The unit is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.

The term unital algebra is also encountered.

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