In mathematics, the concept of an inverse element generalises the concepts of opposite (−x) and reciprocal (1/x) of numbers.

Given an operation denoted here ∗, and an identity element denoted e, if x ∗ y = e, one says that x is a left inverse of y, and that y is a right inverse of x. (An identity element is an element such that x * e = x and e * y = y for all x and y for which the left-hand sides are defined.^[1])

When the operation ∗ is associative, if an element x has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the inverse element or simply the inverse. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an invertible element, also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under addition).

Inverses are commonly used in groups—where every element is invertible, and rings—where invertible elements are also called units. They are also commonly used for operations that are not defined for all possible operands, such as inverse matrices and inverse functions. This has been generalized to category theory, where, by definition, an isomorphism is an invertible morphism.

The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'. This may take its origin from the case of fractions, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of ${\displaystyle {\tfrac {x}{y}}}$ is ${\displaystyle {\tfrac {y}{x}}}$).

Definitions and basic properties

The concepts of inverse element and invertible element are commonly defined for binary operations that are everywhere defined (that is, the operation is defined for any two elements of its domain). However, these concepts are also commonly used with partial operations, that is operations that are not defined everywhere. Common examples are matrix multiplication, function composition and composition of morphisms in a category. It follows that the common definitions of associativity and identity element must be extended to partial operations; this is the object of the first subsections.

In this section, X is a set (possibly a proper class) on which a partial operation (possibly total) is defined, which is denoted with ${\displaystyle *.}$

Associativity

A partial operation is associative if

${\displaystyle x*(y*z)=(x*y)*z}$

for every x, y, z in X for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined.

Examples of non-total associative operations are multiplication of matrices of arbitrary size, and function composition.

Identity elements

Let ${\displaystyle *}$ be a possibly partial associative operation on a set X.

An identity element, or simply an identity is an element e such that

${\displaystyle x*e=x\quad {\text{and}}\quad e*y=y}$

for every x and y for which the left-hand sides of the equalities are defined.

If e and f are two identity elements such that ${\displaystyle e*f}$ is defined, then ${\displaystyle e=f.}$ (This results immediately from the definition, by ${\displaystyle e=e*f=f.}$)

It follows that a total operation has at most one identity element, and if e and f are different identities, then ${\displaystyle e*f}$ is not defined.

For example, in the case of matrix multiplication, there is one n×n identity matrix for every positive integer n, and two identity matrices of different size cannot be multiplied together.

Similarly, identity functions are identity elements for function composition, and the composition of the identity functions of two different sets are not defined.

Left and right inverses

If ${\displaystyle x*y=e,}$ where e is an identity element, one says that x is a left inverse of y, and y is a right inverse of x.

Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on nonnegative integers, which has 0 as additive identity, and 0 is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.

An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the functions from the integers to the integers. The doubling function ${\displaystyle x\mapsto 2x}$ has infinitely many left inverses under function composition, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps n to either ${\displaystyle 2n}$ or ${\displaystyle 2n+1}$ is a right inverse of the function ${\textstyle n\mapsto \left\lfloor {\frac {n}{2}}\right\rfloor ,}$ the floor function that maps n to ${\textstyle {\frac {n}{2}}}$ or ${\textstyle {\frac {n-1}{2}},}$ depending whether n is even or odd.

More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and only if it is surjective.

In category theory, right inverses are also called sections, and left inverses are called retractions.

Inverses

An element is invertible under an operation if it has a left inverse and a right inverse.

In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if l and r are respectively a left inverse and a right inverse of x, then

${\displaystyle l=l*(x*r)=(l*x)*r=r.}$

The inverse of an invertible element is its unique left or right inverse.

If the operation is denoted as an addition, the inverse, or additive inverse, of an element x is denoted ${\displaystyle -x.}$ Otherwise, the inverse of x is generally denoted ${\displaystyle x^{-1},}$ or, in the case of a commutative multiplication ${\textstyle {\frac {1}{x}}.}$ When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in ${\displaystyle x^{*-1}.}$ The notation ${\displaystyle f^{\circ -1}}$ is not commonly used for function composition, since ${\textstyle {\frac {1}{f}}}$ can be used for the multiplicative inverse.

If x and y are invertible, and ${\displaystyle x*y}$ is defined, then ${\displaystyle x*y}$ is invertible, and its inverse is ${\displaystyle y^{-1}x^{-1}.}$

An invertible homomorphism is called an isomorphism. In category theory, an invertible morphism is also called an isomorphism.

In groups

A group is a set with an associative operation that has an identity element, and for which every element has an inverse.

Thus, the inverse is a function from the group to itself that may also be considered as an operation of arity one. It is also an involution, since the inverse of the inverse of an element is the element itself.

A group may act on a set as transformations of this set. In this case, the inverse ${\displaystyle g^{-1}}$ of a group element ${\displaystyle g}$ defines a transformation that is the inverse of the transformation defined by ${\displaystyle g,}$ that is, the transformation that "undoes" the transformation defined by ${\displaystyle g.}$

For example, the Rubik's cube group represents the finite sequences of elementary moves. The inverse of such a sequence is obtained by applying the inverse of each move in the reverse order.

In monoids

A monoid is a set with an associative operation that has an identity element.

The invertible elements in a monoid form a group under monoid operation.

A ring is a monoid for ring multiplication. In this case, the invertible elements are also called units and form the group of units of the ring.

If a monoid is not commutative, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible).

For example, the set of the functions from a set to itself is a monoid under function composition. In this monoid, the invertible elements are the bijective functions; the elements that have left inverses are the injective functions, and those that have right inverses are the surjective functions.

Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the cancellation property (an element x has the cancellation property if ${\displaystyle xy=xz}$ implies ${\displaystyle y=z,}$ and ${\displaystyle yx=zx}$ implies ${\displaystyle y=z}$). This extension of a monoid is allowed by Grothendieck group construction. This is the method that is commonly used for constructing integers from natural numbers, rational numbers from integers and, more generally, the field of fractions of an integral domain, and localizations of commutative rings.

In rings

A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers.

Under addition, a ring is an abelian group, which means that addition is commutative and associative; it has an identity, called the additive identity, and denoted 0; and every element x has an inverse, called its additive inverse and denoted −x. Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.

Under multiplication, a ring is a monoid; this means that multiplication is associative and has an identity called the multiplicative identity and denoted 1. An invertible element for multiplication is called a unit. The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit x is denoted ${\displaystyle x^{-1},}$ or, when the multiplication is commutative, ${\textstyle {\frac {1}{x}}.}$

The additive identity 0 is never a unit, except when the ring is the zero ring, which has 0 as its unique element.

If 0 is the only non-unit, the ring is a field if the multiplication is commutative, or a division ring otherwise.

In a noncommutative ring (that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the linear functions from a infinite-dimensional vector space to itself.

A commutative ring (that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not zero divisors (that is, their product with a nonzero element cannot be 0). This is the process of localization, which produces, in particular, the field of rational numbers from the ring of integers, and, more generally, the field of fractions of an integral domain. Localization is also used with zero divisors, but, in this case the original ring is not a subring of the localisation; instead, it is mapped non-injectively to the localization.

Matrices

Matrix multiplication is commonly defined for matrices over a field, and straightforwardly extended to matrices over rings, rngs and semirings. However, in this section, only matrices over a commutative ring are considered, because of the use of the concept of rank and determinant.

If A is a m×n matrix (that is, a matrix with m rows and n columns), and B is a p×q matrix, the product AB is defined if n = p, and only in this case. An identity matrix, that is, an identity element for matrix multiplication is a square matrix (same number for rows and columns) whose entries of the main diagonal are all equal to 1, and all other entries are 0.

An invertible matrix is an invertible element under matrix multiplication. A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R (that is, is invertible in R. In this case, its inverse matrix can be computed with Cramer's rule.

If R is a field, the determinant is invertible if and only if it is not zero. As the case of fields is more common, one see often invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings. Zdroj:https://en.wikipedia.org?pojem=One-sided_inverse
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