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2005/06/18 | 乘法单位[Multiplicative Identity]
类别(∑〖数学〗)
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发表于 13:25
In a
set
X
equipped with a
binary operation
called a
product
, the multiplicative identity is an element e 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
, 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
or
for all integers
n
> 1.
If
R
is a commutative unit ring, the
constant polynomial
1 is the multiplicative identity of every
polynomial ring
.
In a
Boolean algebra
, if the operation
is considered as a product, the multiplicative identity is the
universal bound
I
. In the
power set
P(S)
of a set
S
, this is the
total set
S
.
The unique element of a
trivial ring
is simultaneously the
additive identity
and
multiplicative identity
.
In a group of maps over a set S (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
S
.
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
K, 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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