Invertible

FORMAL DEFINITION

. If $e$ is an Identity Element of $(S,---)$ and $a---b=e$ , then $a$ is called a left inverse of $b$ and $b$ is called a '''right inverse''' of $a$ . If an element $x$ is both a left inverse and a right inverse of $y$ , then $x$ is called a '''two-sided inverse''', or simply an '''inverse''', of $y$ . An element with a two-sided inverse in $S$ is called '''invertible''' in $S$ .

) can have several left identities or several right identities, it is possible for an element to have several left inverses or several right inverses (but note that their definition above uses a ''two-sided'' identity $e$ ). It can even have several left inverses ''and'' several right inverses.

is Associative then if an element has both a left inverse and a right inverse, they are equal and unique. In this case, the set of (left and right) invertible elements is a Group , called the Group Of Units of $S$ , and denoted by $U(S)$ or $S^---$ .

EXAMPLES

Every Real Number

x

has an Additive Inverse (i.e. an inverse with respect to Addition ) given by

-x

. Every nonzero real number

x

has a Multiplicative Inverse (i.e. an inverse with respect to Multiplication ) given by

rac 1{x}

. By contrast, Zero has no multiplicative inverse.

A Square Matrix

M

with entries in a Field

K

is invertible (in the set of all square matrices of the same size, under Matrix Multiplication ) if and only if its Determinant is different from zero. If the determinant of

M

is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See Invertible Matrix for more.

More generally, a square matrix over a Commutative Ring

R

is invertible Iff its determinant is invertible in

R

.

A function

g

is the left (resp. right) inverse of a function

f

(for Function Composition ), iff

g o f

(resp.

f o g

) is the Identity Function on the Domain (resp. Codomain ) of

f

. In this example, it is very frequent for a function to have a right inverse and no left inverse, or the converse.

SEE ALSO

Additive Inverse

Multiplicative Inverse

Group

Loop

Division Ring

Unit (ring Theory)

	CATEGORIES ABOUT INVERSE ELEMENT

	algebra

	abstract algebra

	binary operations

	inverse element

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Information About

Invertible