Noncommutative unique factorization domain

In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.

• The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a₀ + a₁i + a₂j + a₃k is integral if either all the coefficients a_i are integers or all of them are half-integers.
• All free associative algebras.¹