Multiplicative Identity
In a set 
equipped with
a binary operation 
called a product, the multiplicative identity is an element 
such that
 |
for all 
. It can be, for example, the identity
element of a multiplicative group or the
unit of a unit ring. In both cases it is usually denoted
1. The number 1 is, in fact, the multiplicative identity of the ring
of integers 
and of its extension
rings such as the ring of Gaussian
integers ![Z[i]](/images/equations/MultiplicativeIdentity/Inline6.gif)
, the field
of rational numbers 
, the field
of real numbers 
, and the field
of complex numbers 
. The residue
class 
of number 1 is the multiplicative identity
of the quotient ring 
of 
for all integers
.
If 
is a commutative unit ring, the constant
polynomial 1 is the multiplicative identity of every polynomial
ring ![R[x_1,...,x_n]](/images/equations/MultiplicativeIdentity/Inline15.gif)
.
In a Boolean algebra, if the operation 
is considered
as a product, the multiplicative identity is the universal
bound 
. In the power
set 
of a set 
, this is the total set 
.
The unique element of a trivial ring 
is simultaneously
the additive identity and multiplicative identity.
In a group of maps over a set 
(as, e.g., a transformation group or a symmetric
group), where the product is the map composition, the multiplicative identity
is the identity map on 
.
In the set of 
matrices
with entries in a unit ring, the multiplicative identity
(with respect to matrix multiplication)
is the identity matrix. This is also the multiplicative
identity of the general linear group 
on a field 
, and of all its subgroups.
Not all multiplicative structures have a multiplicative identity. For example, the set of all 
matrices having determinant
equal to zero is closed under multiplication, but this set does not include the identity matrix.
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