Definition:Unital Algebra
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.
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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.