identity element
التعريفات والمعاني
== English ==
=== Noun ===
identity element (plural identity elements)
(algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.
2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,
Let
(
G
,
⋅
)
{\displaystyle (G,\cdot )}
be a group. Then the identity element
e
∈
G
{\displaystyle e\in G}
is unique. […]
Proof. If
e
{\displaystyle e}
and
e
′
{\displaystyle e'}
are both identity elements, then we have
e
e
′
=
e
{\displaystyle ee'=e}
since
e
′
{\displaystyle e'}
is an identity element, and
e
e
′
=
e
′
{\displaystyle ee'=e'}
since
e
{\displaystyle e}
is an identity element. Thus
e
=
e
e
′
=
e
′
{\displaystyle e=ee'=e'}
.
==== Usage notes ====
For binary operation
∗
{\displaystyle *}
defined on a given algebraic structure, an element
i
{\displaystyle i}
is:
a left identity if
i
∗
x
=
x
{\displaystyle i*x=x}
for any
x
{\displaystyle x}
in the structure,
a right identity,
x
∗
i
=
x
{\displaystyle x*i=x}
for any
x
{\displaystyle x}
in the structure,
simply an identity element or (for emphasis) a two-sided identity if both are true.
Where a given structure
M
{\displaystyle M}
is equipped with an operation called addition, the notation
0
M
{\displaystyle 0_{M}}
may be used for the additive identity. Similarly, the notation
1
M
{\displaystyle 1_{M}}
denotes a multiplicative identity.
==== Synonyms ====
(element that when applied with a binary operation leaves any other element unchanged): identity, neutral element
==== Hyponyms ====
(element that when applied with a binary operation leaves any other element unchanged): additive identity, multiplicative identity, zero, zero element
==== Related terms ====
left identity
right identity
==== Translations ====
=== See also ===
unity
=== Further reading ===
Identity matrix on Wikipedia.Wikipedia
Inverse element on Wikipedia.Wikipedia