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Ascending Chain Condition For Principal Ideals
In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbreviated to ACCP) is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant. The counterpart descending chain condition may also be applied to these posets, however there is currently no need for the terminology "DCCP" since such rings are already called left or right perfect rings. (See '' below.) Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and left or right perfect rings. Commutative rings It is well known that a nonzero nonunit in a Noetherian integral domain factors into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form category (mathematics), mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, . If the premise of the lemma does not hold, i.e., is a composite number, its consequent may be either true or false. For example, in the case of , , , composite number 10 divides , but 10 divides neither 4 nor 15. This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's Lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains. Formulations Euclid's lemma is commonly used in the following equivalent form: Euclid's lemma can be generalized as follows from prime numbers to any integers. This is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Finite Ring
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto some proper subset ''B'' of ''A''. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. A simple example is \mathbb, the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number ''n'' to its square ''n''2. Since the set of squares is a proper subset of \mathbb, \mathbb is Dedekind-infinite. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentiet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotence
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyman Bass
Hyman Bass (; born October 5, 1932). The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the c ... References External links *Directory page at University of MichiganAuthor profilein the database zbMATH {{DEFAULTSORT:Bass, Hyman 1932 births 20th-century American Jews Algebraists Columbia University faculty Fellows of the American Academy of Arts and Sciences Fellows of the American Mathematical Society Living people Mathematics educators Members of the United States National Academy of Sciences National Medal of Science laureates Institute for Advanced Study visiting scholars Nicolas Bourbaki Presidents of the American Mathematical Society Academics from Houston Princeton University alumni University of Chicago alumni University of Michigan faculty Mathematicians from Texas 21st-century American Jews ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. Definition The left ''R''-module ''M'' is finitely generated if there exist ''a''1, ''a''2, ..., ''a''''n'' in ''M'' such that for any ''x'' in ''M'', there exist ''r''1, ''r''2, ..., ''r''''n'' in ''R'' with ''x'' = ''r''1''a''1 + ''r''2''a''2 + ... + ''r''''n''''a''''n''. The set is referred to as a genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout Domain
In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French mathematician Étienne Bézout. Examples * All PIDs are Bézout domains. * Examples of Bézout domains that are not PIDs include the ring of entire functions (functions holomorphic on the whole complex plane) and the ring of all algebraic integers. In case of entire functions, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ''a'' has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicatively Closed Subset
In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite products, including the empty product 1.Eisenbud, p. 59. Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring. Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings. A subset ''S'' of a ring ''R'' is called saturated if it is closed under taking divisors: i.e., whenever a product ''xy'' is in ''S'', the elements ''x'' and ''y'' are in ''S'' too. Examples Examples of multiplicative sets include: * the set-theoretic complement of a prime ideal in a commutative ring; * the set , where ''x'' is an element of a ring; * the set of units of a ring; * the set of non-zero-divisors in a ring; * for an ideal ''I''. * the Jor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GCD Domain
In mathematics, a GCD domain is an integral domain ''R'' with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of ''R'' have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non- Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions: Properties Every irreducible element of a GCD domain is prime. A GCD domain is integrally closed, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain. For every pair of elements ''x'', ''y'' of a GCD domain ''R'', a GCD ''d'' of ''x'' and ''y'' and an LCM ''m'' of ''x'' and ''y'' can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general. Definition An element of a commutative ring is said to be prime if it is not the zero element or a unit and whenever divides for some and in , then divides or divides . With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers. Equivalently, an element is prime if, and only if, the principal ideal generated by is a nonzero prime ideal. (Note that in an integral domain, the ideal is a prime ideal, but is an exception in the definition of 'prime element'.) Interest in prime elements comes from the fundamental theorem of arithmetic, which asserts that each non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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