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''neutral element'', ''unit element'', ''zero element''

An element of an algebraic structure $S$, often denoted $1$, $I$ or $e$, with a specific property with respect to a [[binary operation]] $*$:
$$
\forall x \in S \ \ 1 * x = x * 1 = x\ .
$$

Where the binary operation is written additively, the identity element may be denoted $0$ or $O$, and termed a [[zero element]].