Noncommutative unique factorization domain
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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.[1]
References
- "Certain number-theoretic episodes in algebra", R. Sivaramakrishnan; Publisher CRC Press, 2006, ISBN 0824758951
Notes
- ^ "Skew fields: theory of general division rings", Paul Moritz Cohn; Publisher Cambridge University Press, 1995 ISBN 0521432170
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