Square-free element

In mathematics, a square-free element is an element "r" of a unique factorization domain "R" that is not divisible by a non-trivial square. This means that every "s" such that formula_1 is a unit of "R".
Alternate characterizations.
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
Then "r" is square-free if and only if the primes "pi" 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).
Examples.
Common examples of square-free elements include square-free integers and square-free polynomials.