Definition:Unit of Algebra

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Definition

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


The unit of $\left({A_R, \oplus}\right)$, denoted $1_A$, is the identity element of the operation $\oplus$:

$\forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$

It is sometimes referred to as the multiplicative identity of $\left({A_R, \oplus}\right)$.


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


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