|
Krull's Theorem
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice. Variants * For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. * For pseudo-rings, the theorem holds for regular ideals. * An ''apparently'' slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: :::Let ''R'' be a ring, and let ''I'' be a proper ideal of ''R''. Then there is a maximal ideal of ''R'' containing ''I''. :The statement of the original theorem can be obtained by taking ''I'' to be the zero ideal (0). Conversely, applying the original theorem to ''R''/''I'' leads to this result. :To prove the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull's Principal Ideal Theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, ''Krulls Hauptidealsatz'' (from ' ("Principal") + ' + ' ("theorem")). Precisely, if ''R'' is a Noetherian ring and ''I'' is a principal, proper ideal of ''R'', then each minimal prime ideal containing ''I'' has height at most one. This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem. This says that if ''R'' is a Noetherian ring and ''I'' is a proper ideal generated by ''n'' elements of ''R'', then each minimal prime over ''I'' has height at most ''n''. The converse is also true: if a prime ideal has height ''n'', then it is a minimal prime ideal over an ideal generated by ''n'' elements. The principal ideal theorem and the generalization, the height theorem, both follow from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets (S_i as a nonempty set indexed with i), there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. The axiom of choice is equivalent to the statement that every partition has a transversal. In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, Nigel Hitchin, and Thomas Schick. Currently, the managing editor of Mathematische Annalen is Yoshikazu Giga (University of Tokyo). Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947, the journal briefly ceased publication. References External links''Mathematische Annalen''homepage a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Ideal
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An '' additive identity'' is the identity element in an additive group or monoid. It corresponds to the element 0 such that for all x in the group, . Some examples of additive identity include: * The zero vector under vector addition: the vector whose components are all 0; in a normed vector space its norm (length) is also 0. Often denoted as \mathbf or \vec. * The zero function or zero map defined by , under pointwise addition * The ''empty set'' under set union * An '' empty sum'' or ''empty coproduct'' * An ''initial object'' in a category (an empty coproduct, and so an identity under coproducts) Absorbing elements An '' absorbing element'' in a multiplicative semigroup or semiring generalises the property . Examples include: *The ''empty set' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Ideal
In mathematics, and more specifically in ring theory, 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Ideal
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal (ring theory), ideal \mathfrak in a (possibly) non-unital ring ''A'' is said to be regular (or modular) if there exists an element ''e'' in ''A'' such that ex - x \in \mathfrak for every x \in A. In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor. This article will use "regular element ideal" to help distinguish this type of ideal. A two-sided ideal \mathfrak of a ring ''R'' can also be called a (von Neumann) regular ideal if for each element ''x'' of \mathfrak there exists a ''y'' in \mathfrak such that ''xyx''=''x''. Finally, regular ideal has been used to refer to an ideal ''J'' of a ring ''R'' such that the quotient ring ''R''/''J'' is von Neumann regular ring.Burton, D.M. (1970) ''A first course in rings and ideals.'' Addison-Wesley. Reading, Massachusetts . This article will use "quotient von Neumann regular" to refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-ring
In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: * A rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity. * A set ''R'' with two binary operations + and ⋅ such that is an abelian group with identity 0, and and for all ''a'', ''b'', ''c'' in ''R''. * An abelian group equipped with a subgroup ''B'' and a multiplication making ''B'' a ring and ''A'' a ''B''- module. None of these definitions are equivalent, so it is best to avoid the term "pseudo-ring" or to clarify which meaning is intended. See also * Semiring In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ... – an algebraic structure similar to a ring, but without the requirement that each ele ... [...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 an 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 shorthand 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zorn's Lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element. The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure. Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF ( Zermelo–Fraenkel set theory without th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfinite Induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then is also true. Then transfinite induction tells us that is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that is true. * Successor case: Prove that for any successor ordinal , follows from (and, if necessary, for all ). * Limit case: Prove that for any [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
