In mathematics, an element r of a unique factorization domain R is called square-free, if it is not divisible by a non-trivial square. That is, if every s such that $s^2\backslash mid\; r$ is a unit of R.

Square-free elements may be also characterized using their prime decomposition. The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements

$r=p\_1p\_2\backslash cdots\; p\_n$

Then r is square-free if and only if the primes p_i are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it divisible by a square number).

Common examples of square-free elements include square-free integers and square-free polynomials.

Ring theory