In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a noncommutative unique factorization domain is a
noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...
with the
unique factorization property.
Examples
*The
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of
Hurwitz quaternion In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are ''either'' all integers ''or'' all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz qua ...
s, also known as integral
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s. A quaternion ''a'' = ''a''
0 + ''a''
1i + ''a''
2j + ''a''
3k is integral if either all the coefficients ''a''
i are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s or all of them are
half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
s.
* All
free associative algebras.
[Cohn, p. 329]
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,
Notes
Ring theory
Number theory
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