Editing Square-free integer (section)

==Equivalent characterizations==
A positive integer <math>n</math> is square-free if and only if in the [[canonical representation of a positive integer|prime factorization]] of <math>n</math>, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime [[divisor|factor]] <math>p</math> of <math>n</math>, the prime <math>p</math> does not evenly divide&nbsp;<math>n/p</math>. Also <math>n</math> is square-free if and only if in every factorization <math>n=ab</math>, the factors <math>a</math> and <math>b</math> are [[coprime]].  An immediate result of this definition is that all prime numbers are square-free.

A positive integer <math>n</math> is square-free if and only if all [[abelian group]]s of [[order (group theory)|order]] <math>n</math> are [[group isomorphism|isomorphic]], which is the case if and only if any such group is [[cyclic group|cyclic]]. This follows from the classification of [[finitely generated abelian group]]s.

A integer <math>n</math> is square-free if and only if the [[factor ring]] <math>\mathbb{Z}/n\mathbb{Z}</math> (see [[modular arithmetic]]) is a [[product of rings|product]] of [[field (mathematics)|field]]s. This follows from the [[Chinese remainder theorem]] and the fact that a ring of the form <math>\mathbb{Z}/k\mathbb{Z}</math> is a field if and only if <math>k</math> is prime.

For every positive integer <math>n</math>, the set of all positive divisors of <math>n</math> becomes a [[partially ordered set]] if we use [[divisor|divisibility]] as the order relation. This partially ordered set is always a [[distributive lattice]]. It is a [[Boolean algebra (structure)|Boolean algebra]] if and only if <math>n</math> is square-free.

A positive integer <math>n</math> is square-free [[if and only if]] <math>\mu(n)\ne 0</math>, where <math>\mu</math> denotes the [[Möbius function]].