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.


Sources