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Zorn Ring
In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent ''x'' there exists an element ''y'' such that ''xy'' is a non-zero idempotent . named them after Max August Zorn, who studied a similar condition in . For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(''R'') is a nil ideal and every right ideal of ''R'' which is not contained in J(''R'') contains a nonzero idempotent. Replacing "right ideal" with "left ideal" yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular ring In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the element ...s are all examples of associative Zorn rings. References * * * * Non-associative algebras Ring t ... [...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] |
Alternative Ring
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by : ,y,z= (xy)z - x(yz). By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent toSchafer (1995) p. 27 : ,x,y= 0 : ,x,x= 0. Both of these identities together imply that : ,y,x= , x, x+ , y, x- , x+y, x+y= , x+y, -y= , x, -y- , y, y= 0 for all x and y. This is equivalent to the ''flexible identity''Schafer (1995) p. 28 :(xy)x = x(yx). The associator of an alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. * Assume that two elements a and b in a ring R satisfy ab=0. Then the element c=ba is nilpotent as \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ 0 & 0 \end. Here AB=0 and BA=B. *By definition, any element of a nilsemigroup is nilpotent. Properties No nilpotent element c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent Element (ring Theory)
In ring theory, a branch of abstract algebra, an idempotent element or simply idempotent of a ring is an element ''a'' such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer ''n''. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication. Examples Quotients of Z One may consider the ring of integers modulo ''n'' where ''n'' is squarefree. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo ''p'' where ''p'' is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be 0 and 1. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max August Zorn
Max August Zorn (; June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a method used in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, and ordered sets amongst others. Zorn's lemma was first postulated by Kazimierz Kuratowski in 1922, and then independently by Zorn in 1935. Life and career Zorn was born in Krefeld, Germany. He attended the University of Hamburg. He received his PhD in April 1930 for a thesis on alternative algebras. He published his findings in ''Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg''. Zorn showed that split-octonions could be represented by a mixed-style of matrices called Zorn's vector-matrix algebra. Max Zorn was appointed to an assistant position at the University of Halle. However, he did not have the opportunity to work there for long as he was forced to leave Germany ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Rings
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 pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobson Radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or \operatorname(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in . The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma. Definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nil Ideal
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture. Commutative rings In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving nilpotent elements and sums of nilpotent elements are both nilpotent. This is because if ''a'' and ''b'' are nilpotent elements of ''R'' with ''a''n=0 and ''b''m=0, and r is any element of R, then (''a''·''r'')n = ''a''n·''r''n = 0, and by the binomial theorem, (''a''+''b'')m+n=0. Therefore, the set of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Ring
In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book. A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric. Perfect ring Definitions The following equivalent definitions of a left perfect ring ''R'' are found in Aderson and Fuller: * Every left ''R'' module has a projective cover. * ''R''/J(''R'') is semisimple and J(''R'') is left T-nilpotent (that is, for every infinite sequence of elements of J(''R'') there is an ''n'' such that the product of first ''n'' terms are zero), where J(''R'') is the Jacobson radical of ''R''. * (Bass' Theorem P) ''R'' satisfies the descending chain condition on principal rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopkins–Levitzki Theorem
In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in Module (mathematics), modules over semiprimary rings. A ring ''R'' (with 1) is called semiprimary if ''R''/''J''(''R'') is semisimple algebra, semisimple and ''J''(''R'') is a nilpotent ideal, where ''J''(''R'') denotes the Jacobson radical. The theorem states that if ''R'' is a semiprimary ring and ''M'' is an ''R'' module, the three module conditions noetherian module, Noetherian, artinian module, Artinian and "has a composition series" are equivalent. Without the semiprimary condition, the only true implication is that if ''M'' has a composition series, then ''M'' is both Noetherian and Artinian. The theorem takes its current form from a paper by Charles Hopkins and a paper by Jacob Levitzki, both in 1939. For this reason it is often cited as the Hopkins–Levitzki theorem. However Yasuo Akizuki is sometim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
