In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two It leaves other elements unchanged when combined with them. This is used for groups and related concepts. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure.

The term identity element is often shortened to identity when there is no possibility of confusion; we do so in this article.

Let (S,*) be a set S with a binary operation * on it (known as a magma). In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as rings. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The multiplicative identity is often called the unit in the latter context, where, unfortunately, a unit is also sometimes used to mean an element with a multiplicative inverse. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

Examples

set	operation	identity
real numbers	+ (addition)	0
real numbers	• (multiplication)	1
real numbers	ab (exponentiation)	1 (right identity only)
m-by-n matrices	+ (addition)	zero matrix
n-by-n square matrices	• (multiplication)	identity matrix
all functions from a set M to itself	∘ (function composition)	identity map
all functions from a set M to itself	* (convolution)	δ (Dirac delta)
character strings, lists	concatenation	empty string, empty list
extended real numbers	minimum/infimum	+∞
extended real numbers	maximum/supremum	−∞
subsets of a set M	∩ (intersection)	M
sets	∪ (union)	{ } (empty set)
boolean logic	∧ (logical and)	⊤ (truth)
boolean logic	∨ (logical or)	⊥ (falsity)
compact surfaces	# (connected sum)	S²
only two elements {e, f}	* defined by e * e = f * e = e and f * f = e * f = f	both e and f are left identities, but there is no right or two-sided identity

As the last example shows, it is possible for (S,*) to have several left identities. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the real numbers may be described informally in several different ways Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, particularly Linear algebra, a zero matrix is a matrix with all its entries being zero. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. In Computer science and Formal language theory, the empty string is the unique string of length Zero. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of See also Classification of manifolds#Point-set In Mathematics, a closed manifold is type of Topological space, namely a compact In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l * r = r. In particular, there can never be more than one two-sided identity. If there were two, e and f, then e * f would have to be equal to both e and f.

It is also quite possible for an algebra to have no identity element. The most common examples of this are the dot product and cross product of vectors. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In the former case the lack of an identity element is related to the fact that the elements multiplied are vectors but the product is a scalar. With cross products the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied - so that it is not possible to obtain a vector in the same direction as the original. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

See also

• Inverse element
• Additive inverse
• Monoid
• Unital
• Quasigroup

In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division

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