An additive group is an abelian group whose operation is written as addition, with identity and inverse of written .
A nonzero finitely generated subgroup of the additive rational numbers is an infinite cyclic group. Put the generators over a common positive denominator. Their integer numerators generate the ideal given by their greatest common divisor, and Bezout identity expresses that divisor as an integer combination of them. Dividing back by the denominator gives a single generator.
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An additive group is a mathematical structure that consists of a set equipped with an operation (usually referred to as addition) that satisfies four key properties: closure, associativity, the existence of an identity element, and the existence of inverses.