Identitätselements

Identitätselement is a term used in abstract algebra to describe a specific type of element within a set on which a binary operation is defined. For a binary operation denoted by *, an identity element, often symbolized as 'e', possesses the property that when it is combined with any element 'a' from the set, the result is always the element 'a' itself. This property can be expressed as e * a = a and a * e = a for all elements 'a' in the set.

The existence of an identity element is a crucial characteristic for certain algebraic structures. For instance, in the context of a group, the presence of an identity element is one of the defining axioms. The set of integers with the operation of addition has an identity element of 0, because for any integer 'n', n + 0 = n and 0 + n = n. Similarly, the set of non-zero rational numbers with the operation of multiplication has an identity element of 1, as for any non-zero rational number 'q', q * 1 = q and 1 * q = q.

It is important to note that not all sets with binary operations have an identity element. For example, the set of natural numbers {1, 2, 3, ...} with the operation of addition does not have an identity element because there is no natural number that, when added to another natural number, leaves the second number unchanged. The identity element, if it exists, is always unique within a given set and operation.