Multiplicative Identity

In a set 👁 X
equipped with a binary operation 👁 ·
called a product, the multiplicative identity is an element 👁 e
such that

for all 👁 x in X
. 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 👁 Z
and of its extension rings such as the ring of Gaussian integers 👁 Z[i]
, the field of rational numbers 👁 Q
, the field of real numbers 👁 R
, and the field of complex numbers 👁 C
. The residue class 👁 1^_
of number 1 is the multiplicative identity of the quotient ring 👁 Z_n
of 👁 Z
for all integers 👁 n>1
.

If 👁 R
is a commutative unit ring, the constant polynomial 1 is the multiplicative identity of every polynomial ring 👁 R[x_1,...,x_n]
.

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 👁 n×n
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 👁 GL(n,K)
on a field 👁 K
, and of all its subgroups.

Not all multiplicative structures have a multiplicative identity. For example, the set of all 👁 n×n
matrices having determinant equal to zero is closed under multiplication, but this set does not include the identity matrix.

See also

Additive Identity, Multiplicative Inverse

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Multiplicative Identity." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MultiplicativeIdentity.html

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