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 , 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 .

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.