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Glossary Of Ring Theory
Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. For the items in commutative algebra (the theory of commutative rings), see '' Glossary of commutative algebra''. For ring-theoretic concepts in the language of modules, see also '' Glossary of module theory''. For specific types of algebras, see also: ''Glossary of field theory Field theory (mathematics), Field theory is the branch of mathematics in which field (mathematics), fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) De ...'' and '' Glossary of Lie groups and Lie algebras''. Since, currently, there is no glossary on not-necessarily-associative algebra structures in general, this glossary includes some concepts that do not need associativity; e.g., a derivation. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bidimension Of An Associative Algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of ''K''). The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a module or vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a commutative ring ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring. In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Subring
A characteristic is a distinguishing feature of a person or thing. It may refer to: Computing * Characteristic (biased exponent), an ambiguous term formerly used by some authors to specify some type of exponent of a floating point number * Characteristic (significand), an ambiguous term formerly used by some authors to specify the significand of a floating point number Science *''I–V'' or current–voltage characteristic, the current in a circuit as a function of the applied voltage *Receiver operating characteristic Mathematics * Characteristic (algebra) of a ring, the smallest common cycle length of the ring's addition operation * Characteristic (logarithm), integer part of a common logarithm * Characteristic function, usually the indicator function of a subset, though the term has other meanings in specific domains * Characteristic polynomial, a polynomial associated with a square matrix in linear algebra * Characteristic subgroup, a subgroup that is invariant under all autom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Centralizer Theorem
In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring ''S'' of a ring ''R'', denoted C''R''(''S'') in this article. It is always the case that C''R''(C''R''(''S'')) contains ''S'', and a double centralizer theorem gives conditions on ''R'' and ''S'' that guarantee that C''R''(C''R''(''S'')) is ''equal'' to ''S''. Statements of the theorem Motivation The centralizer of a subring ''S'' of ''R'' is given by :\mathrm_R(S)=\.\, Clearly C''R''(C''R''(''S'')) ⊇ ''S'', but it is not always the case that one can say the two sets are equal. The double centralizer theorems give conditions under which one can conclude that equality occurs. There is another special case of interest. Let ''M'' be a right ''R'' module and give ''M'' the natural left ''E''-module structure, where ''E'' is End(''M''), the ring of endomorphisms of the abelian group ''M''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Centralizer
Double, The Double or Dubble may refer to: Mathematics and computing * Multiplication by 2 * Double precision, a floating-point representation of numbers that is typically 64 bits in length * A double number of the form x+yj, where j^2=+1 * A 2-tuple, or ordered list of two elements, commonly called an ordered pair, denoted (a,b) * Double (manifold), in topology Food and drink * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doubling cube, in backgammon ** Double, doubling a blackjack bet in a favorable situation ** Double, a bet offered by UK bookmakers which combines two selections * Double, villain in the video game ''Mega Man X4'' * A kart racing game '' Mario Kart: Double Dash'' * An arcade action game ''Double Dragon'' Sports * Double (association football), the act of a winning a division and primary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centralizer (ring Theory)
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of elements g\in G such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and norma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Simple Algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or '' Brauer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Algebra
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(''R''); 'Z' stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples * The center of a commutative ring ''R'' is ''R'' itself. * The center of a skew-field is a field. * The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. * Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then . * The center of the universal enveloping algebra of a Lie algebra plays an important role in the represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center (ring Theory)
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(''R''); 'Z' stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples * The center of a commutative ring ''R'' is ''R'' itself. * The center of a skew-field is a field. * The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. * Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then . * The center of the universal enveloping algebra of a Lie algebra In mathematics, a Lie algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subring
In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all subsets of a ring are rings. Definition A subring of a ring is a subset of that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of . Equivalently, is a subring if and only if it contains the multiplicative identity of , and is closed under multiplication and subtraction. This is sometimes known as the ''subring test''. Variations Some mathematicians define rings without requiring the existence of a multiplicative identity (see '). In this case, a subring of is a subset of that is a ring for the operations of (this does imply it contains the additive identity of ). This alternate definition gives a strictly weaker condition, even for rings that do have a mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper. As a concrete category The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor :''U'' : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :''F'' : Set → Ring which assigns to each set ''X'' the free ring generated by ''X''. One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |