Identity element

In algebra, an identity element or neutral element with respect to a binary operation is an element which leaves the other operand unchanged, generalising the concept of zero with respect to addition or one with respect to multiplication.

Formally, let $$\star$$ be a binary operation on a set X. An element I of X is an identity for $$\star$$ if


 * $$I \star x = x = x \star I \,$$

holds for all x in X. An identity element, if it exists, is unique.

Examples

 * Existence of an identity element is one of the properties of a group or monoid.
 * An identity matrix is the identity element for matrix multiplication.