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Identity element
Identity element
Identity element
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In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied.[1][2] For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity)[3] when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

Definitions

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Let (S, ∗) be a set S equipped with a binary operation ∗. Then an element e of S is called a left identity if e ∗ s = s for all s in S, and a right identity if s ∗ e = s for all s in S.[4] If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.[5][6][7][8][9]

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).[3] These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a group for example, the identity element is sometimes simply denoted by the symbol . The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity).[10][11][12] This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit.[13][14]

Examples

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Set Operation Identity
Real numbers, complex numbers + (addition) 0
· (multiplication) 1
Positive integers Least common multiple 1
Non-negative integers Greatest common divisor 0 (under most definitions of GCD)
Vectors Vector addition Zero vector
Scalar multiplication 1
m-by-n matrices Matrix addition Zero matrix
n-by-n square matrices Matrix multiplication In (identity matrix)
m-by-n matrices ○ (Hadamard product) Jm, n (matrix of ones)
All functions from a set, M, to itself ∘ (function composition) Identity function
All distributions on a group, G ∗ (convolution) δ (Dirac delta)
Extended real numbers Minimum/infimum +∞
Maximum/supremum −∞
Subsets of a set M ∩ (intersection) M
∪ (union) ∅ (empty set)
Strings, lists Concatenation Empty string, empty list
A Boolean algebra (conjuction) (truth)
(equivalence) (truth)
(disjunction) (falsity)
(nonequivalence) (falsity)
Knots Knot sum Unknot
Compact surfaces # (connected sum) S2
Groups Direct product Trivial group
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 identity
and no two-sided identity
Homogeneous relations on a set X Relative product Identity relation
Relational algebra Natural join (⨝) The unique relation degree zero and cardinality one

Properties

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In the example S = {e,f} with the equalities given, S is a semigroup. It demonstrates the possibility for (S, ∗) to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in 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, say e and f, then e ∗ f would have to be equal to both e and f.

It is also quite possible for (S, ∗) to have no identity element,[15] such as the case of even integers under the multiplication operation.[3] Another common example is the cross product of vectors, where 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. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive semigroup of positive natural numbers.

See also

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Notes and references

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Bibliography

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Identity element

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from Grokipedia
In , an identity element (also called a neutral element) for a on a set is an element ee that, when combined with any other element aa in the set using the operation, yields aa itself. Formally, ee satisfies a∗e=e∗a=aa * e = e * a = a for all aa in the set, where ∗* denotes the . If an exists for a on a nonempty set, it is unique. To see this, suppose ee and e′e' both serve as identities; then e∗e′=e′e * e' = e' (since ee is an identity) and e∗e′=ee * e' = e (since e′e' is an identity), implying e=e′e = e'. Not every has an identity—for instance, the operation x∗y=1+xyx * y = 1 + xy on the integers lacks one. Common examples include 0 as the for the under , since n+0=0+n=nn + 0 = 0 + n = n for any nn, and 1 as the multiplicative identity under , since n×1=1×n=nn \times 1 = 1 \times n = n. Another case is the operation x∗y=x+y+1x * y = x + y + 1 on the , where -1 acts as the identity. The identity element plays a foundational role in algebraic structures such as monoids and groups, where its existence is a defining that enables concepts like inverses and ensures operational consistency.

Definitions

Basic Definition

In mathematics, particularly in , an identity element is defined within the context of a set equipped with a . Consider a non-empty set SS and a ∗* on SS, which is a function from the S×SS \times S to SS, meaning that for every pair of elements a,b∈Sa, b \in S, the result a∗ba * b is also in SS (a property known as closure). This setup forms a binary structure ⟨S,∗⟩\langle S, * \rangle, where the operation is well-defined but no further properties, such as associativity, are assumed at this stage. An element e∈Se \in S is called an identity element (or neutral element) for the binary structure ⟨S,∗⟩\langle S, * \rangle if it satisfies the two-sided condition: for all a∈Sa \in S, a∗e=e∗a=a.a * e = e * a = a. This formal requirement ensures that ee interacts with every element in a way that preserves the original element under the operation from either side, establishing the standard definition of a two-sided identity. The identity element serves as a neutral counterpart to the operation, leaving other elements unchanged when combined with them, which facilitates the of more complex algebraic behaviors in structures that possess such an element. While not every binary structure contains an identity, its presence defines a unital magma.

Left and Right Identities

In algebraic structures such as magmas, a left identity is an element ee in the set SS with respect to a ∗* such that e∗a=ae * a = a for all a∈Sa \in S, with no requirement that a∗e=aa * e = a. Similarly, a right identity is an element e∈Se \in S such that a∗e=aa * e = a for all a∈Sa \in S, without any condition on e∗ae * a. An element that serves as both a left identity and a right identity is termed a two-sided identity, satisfying e∗a=a∗e=ae * a = a * e = a for all a∈Sa \in S; however, the existence of a one-sided identity does not guarantee the other side, allowing for cases where left or right identities appear independently in non-standard operations. For instance, consider the infinite set of natural numbers with the operation x∗y=yx * y = y, which is associative and renders every element a left identity (x∗y=yx * y = y) but admits no right identity.

Properties

Uniqueness

In algebraic structures equipped with a , the identity element, if it exists, is unique. To see this, suppose ee and ff are both identity elements for the operation ⋅\cdot on a set SS, meaning a⋅e=e⋅a=aa \cdot e = e \cdot a = a and a⋅f=f⋅a=aa \cdot f = f \cdot a = a for all a∈Sa \in S. Then, e=e⋅f=fe = e \cdot f = f, where the first equality uses the right-identity property of ff and the second uses the left-identity property of ee. This proof follows directly from the definition of a two-sided identity and requires no additional axioms beyond the existence of the operation itself. It applies to any (a set with a ) that possesses such an element. For one-sided identities, uniqueness holds under certain conditions. Specifically, if a has both a left identity ll (satisfying l⋅m=ml \cdot m = m for all m∈Sm \in S) and a right identity rr (satisfying m⋅r=mm \cdot r = m for all m∈Sm \in S), then l=l⋅r=rl = l \cdot r = r, making it a unique two-sided identity. However, a left identity alone may not be unique. The uniqueness of the identity element has significant implications for classifying algebraic structures. For example, in the definition of a —a with an associative operation and an identity—the presence of the identity guarantees exactly one such element, facilitating the study of further properties like inverses and homomorphisms.

Behavior in Algebraic Structures

In a (M,⋅)(M, \cdot), the identity element ee is required by as the unique neutral element that satisfies e⋅a=a⋅e=ae \cdot a = a \cdot e = a for all a∈Ma \in M, with the ⋅\cdot being associative. This structure ensures that ee acts as a fixed point under the operation, enabling consistent composition without altering elements. The associativity of ⋅\cdot guarantees that ee interacts uniformly in any parenthesized expression, preserving the monoid's operational integrity. Groups extend monoids by incorporating inverses relative to the identity: for every a∈Ga \in G, there exists a−1∈Ga^{-1} \in G such that a⋅a−1=a−1⋅a=ea \cdot a^{-1} = a^{-1} \cdot a = e, where (G,⋅)(G, \cdot) is associative with ee as the neutral element. This inverse property ties ee directly to solvability, as equations like a⋅x=ea \cdot x = e have unique solutions x=a−1x = a^{-1}, facilitating cancellation and the resolution of operational equations within the group. In rings (R,+,⋅)(R, +, \cdot), distinct identities exist for each operation: the additive identity $0satisfiessatisfiesa + 0 = 0 + a = aforallfor alla \in R, forming an abelian group under addition, while a multiplicative identity $1 (in unital rings) satisfies a⋅1=1⋅a=aa \cdot 1 = 1 \cdot a = a. Distributivity (a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c and (b+c)⋅a=b⋅a+c⋅a(b + c) \cdot a = b \cdot a + c \cdot a) links the operations but leaves the identities' neutral roles unchanged. The 's presence enables between structures: a homomorphism f:(M,⋅)→(N,⋆)f: (M, \cdot) \to (N, \star) preserves the operation (f(a⋅b)=f(a)⋆f(b)f(a \cdot b) = f(a) \star f(b)) and maps eMe_M to eNe_N, similarly for group and ring homomorphisms which also preserve inverses or distributivity as applicable. This preservation allows identities to serve as anchors for mapping structural properties, such as isomorphisms or quotients. In non-unital structures like semigroups, no identity is required, consisting solely of an associative operation without a neutral element, contrasting with and highlighting the identity's optional yet foundational role in more complete algebraic frameworks.

Examples

In Numerical Operations

In the context of numerical operations on the real numbers, the additive identity element is 0, meaning that for any real number aa, a+0=0+a=aa + 0 = 0 + a = a. This property holds for integers as well, where addition forms an abelian group with 0 as the unique identity. For multiplication on the real numbers, the identity element is 1, such that a×1=1×a=aa \times 1 = 1 \times a = a for any real number aa. However, in the multiplicative structure of the real numbers excluding zero, which forms a group, every non-zero element has an inverse, but zero itself lacks a multiplicative inverse in the full set of reals. Subtraction on the real numbers does not possess a global two-sided , as there is no number ee satisfying a−e=e−a=aa - e = e - a = a for all aa; while acts as a right identity since a−[0](/page/0)=aa - [0](/page/0) = a, it fails as a left identity because 0−a=−a≠a0 - a = -a \neq a unless a=[0](/page/0)a = [0](/page/0)./05%3A_Sample_Topics/5.02%3A_Abstract_Algebra-_commutative_groups) Similarly, division lacks a global identity due to domain restrictions ( is undefined) and the absence of a two-sided neutral element; 1 serves only as a right identity, as a/1=aa / 1 = a but 1/a≠a1 / a \neq a for a≠1a \neq 1. In vector spaces over the real numbers, the zero vector 0⃗\vec{0}
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