identity element

Noun

 * 1)  An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.
 * 2) * 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,
 * Let $$(G,\cdot)$$ be a group. Then the identity element $$e\in G$$ is unique.
 * Proof. If $$e$$ and $$e'$$ are both identity elements, then we have $$ee' = e$$ since $$e'$$ is an identity element, and $$ee' = e'$$ since $$e$$ is an identity element. Thus
 * $$e = ee' = e'$$.
 * $$e = ee' = e'$$.

Usage notes
For binary operation $$*$$ defined on a given algebraic structure, an element $$i$$ is:
 * 1) a  if $$i * x = x$$ for any $$x$$ in the structure,
 * a, $$x * i = x$$ for any $$x$$ in the structure,
 * 1) simply an identity element or (for emphasis) a two-sided identity if both are true.

Where a given structure $$M$$ is equipped with an operation called addition, the notation $$0_M$$ may be used for the. Similarly, the notation $$1_M$$ denotes a.

Translations

 * Dutch:
 * Finnish: identiteettialkio
 * Hebrew: איבר יחידה
 * Hungarian:
 * Irish: ball ionannais
 * Italian: elemento neutro
 * Maori: tūmau
 * Russian: ,
 * Spanish: elemento neutro
 * Swedish: