Noncommutative unique factorization domain

In mathematics, the noncommutative unique factorization domain is the noncommutative counterpart of the commutative or classical unique factorization domain (UFD).

Example

• The ring of integral [[quaternions]. If the coefficients a₀, a₁, a₂, a₃ are integers or halves of odd integers of a rational quaternion a = a₀ + a₁i + a₂j + a₃k then the quaternion is integral.

Theorem

Let D be a noncommutative integral domain. D is a unique factorization domain whenever for each nonzero nonunit a D, the set L ( aD , D ) of principal right ideals between D and aD forms a modular lattice of finite length, as a sublattice of the lattice of all ideals of D.

References

• "Certain number-theoretic episodes in algebra", R. Sivaramakrishnan; Publisher CRC Press, 2006, ISBN 0824758951