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Noncommutative Unique Factorization Domain
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'' = ''a''0 + ''a''1i + ''a''2j + ''a''3k is integral if either all the coefficients ''a''i are integers 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 {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term ''noncommutative ring'' is used instead of ''ring'' to refer to a unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ''ring'' is used as a shortcut for ''commutative ring''. Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise. Examples Some examples of noncommutative rings: * The ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Factorization Domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units. Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: Definition Formally, a unique factorization domain is defined to be an integral domain ''R'' in which every non-zero element ''x'' of ''R'' can be written as a product (an empty product if ''x'' is a unit) of irreducible elements ''p''i of ''R'' and a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 quaternions is :H = \left\. That is, either ''a'', ''b'', ''c'', ''d'' are all integers, or they are all half-integers. ''H'' is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by . A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions :L = \left\ forms a subring of the Hurwitz quaternions ''H''. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder. Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings. Struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 such as 1 (being half the integer 2). A name such as "integer-plus-half" may be more accurate, but even though not literally true, "half integer" is the conventional term. Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient. Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the dyadic rationals (numbers produced by dividing an integer by a power of two). Notation and algebraic structure The set of all half-integers is often denoted :\mathbb Z + \tfrac \quad = \quad \left( \tfrac \mathbb Z \right) \smallsetminus \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. Definition For ''R'' a commutative ring, the free (associative, unital) algebra on ''n'' indeterminates is the free ''R''-module with a basis consisting of all words over the alphabet (including the empty word, which is the unit of the free algebra). This ''R''-module becomes an ''R''-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words: :\left(X_X_ \cdots X_\right) \cdot \left(X_X_ \cdots X_\right) = X_X_ \cdots X_X_X_ \cdots X_, and the product of two arbitrary ''R''-module elements is thus uniquely determined (because the multiplication in an ''R''-algebra must be ''R''-bilinear). This ''R''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Theory
In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological algebra, homological properties and Polynomial identity ring, polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of ''commutative algebra'', a major area of modern mathematics. Because these three fields (algebraic geometry, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
