In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
Examples The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a0 + a1i + a2j + a3k is integral if either all the coefficients ai are integers or all of them are half-integers. All free associative algebras.
References P. M. Cohn, "Noncommutative unique factorization domains", Transactions of the American Mathematical Society 109:2:313-331 (1963). full text R. Sivaramakrishnan, Certain number-theoretic episodes in algebra, CRC Press, 2006, ISBN 0-8247-5895-1
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