Inverse element

An inverse element combines with a given element under a binary operation to produce that operation’s identity element. Depending on the operation, inverses may be left-sided, right-sided, or two-sided.

Binary operation: A rule that combines two elements of a set to produce another element. An inverse is defined relative to a particular binary operation.
Additive inverse: The number that sums with a given number to produce zero. It is the inverse under addition, whose identity is zero.
Cancellation property: A property allowing equal results of an operation to imply equality of the corresponding inputs. Cancellation may hold even when an element has no inverse.
Modular arithmetic: Arithmetic on residue classes, where numbers differing by a modulus are treated as equivalent. Multiplicative inverses modulo an integer exist precisely for residues coprime to it.
Identity element: An element that leaves every element unchanged when combined with it under a binary operation. The inverse is defined by combining to this element.
Multiplicative inverse: The number that multiplies with a given nonzero number to produce one. It is the inverse under multiplication, whose identity is one.
Idempotent element: An element that remains unchanged when combined with itself under an operation. Idempotence concerns self-combination, not combination to an identity.
Cryptography: Methods for securing information through mathematical transformations and secret keys. Several cryptographic operations rely on inverses in finite groups or modular arithmetic.
Associativity: A property of an operation whereby grouping a sequence of operations does not change its result. Associativity helps establish familiar laws for two-sided inverses.
Left inverse: An element that combines on the left with another element to produce the identity. Noncommutative operations can have inverses on only this side.

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