|
Elementary Divisors
In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a PID and M a finitely generated R-module, then ''M'' is isomorphic to a finite sum of the form ::M\cong R^r\oplus \bigoplus_^l R/(q_i) \qquad\textr,l\geq0 :where the (q_i) are nonzero primary ideals. The list of primary ideals is unique up to order (but a given ideal may be present more than once, so the list represents a multiset of primary ideals); the elements q_i are unique only up to associatedness, and are called the ''elementary divisors''. Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers q_i=p_i^ of irreducible elements. The nonnegative integer r is called the ''free rank'' or ''Betti number'' of the module M. The module is determined up to isomorphism by specifying its free rank , and for class of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form . Principal ideal domains are noetherian, they are integra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields. Statement When a vector space over a field ''F'' has a finite generating set, then one may extract from it a basis consisting of a finite number ''n'' of vectors, and the space is therefore isomorphic to ''F''''n''. The corresponding statement with the ''F'' generalized to a principal ideal domain ''R'' is no longer true, since a basis for a finitely generated module over ''R'' might not exist. However such a module is still isomorphic to a quotient of some module ''Rn'' with '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form . Principal ideal domains are noetherian, they are integra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Ideal
In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n'' > 0. For example, in the ring of integers Z, (''p''''n'') is a primary ideal if ''p'' is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity. Examples and properties * The definition can be rephrased in a more symmetric manner: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements: * The set contains only elements and , each having multiplicity 1 when is seen as a multiset. * In the multiset , the element has multiplicity 2, and has multiplicity 1. * In the multiset , and both have multiplicity 3. These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, order does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associatedness
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 entire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Factors
The invariant factors of a module (mathematics), module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a Principal ideal domain, PID and M a Finitely-generated module, finitely generated R-module, then :M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m) for some integer r\geq0 and a (possibly empty) list of nonzero elements a_1,\ldots,a_m\in R for which a_1 \mid a_2 \mid \cdots \mid a_m. The nonnegative integer r is called the ''free rank'' or ''Betti number'' of the module M, while a_1,\ldots,a_m are the ''invariant factors'' of M and are unique up to associatedness. The invariant factors of a Matrix (mathematics), matrix over a PID occur in the Smith normal form and provide a means of computing the structure of a module from a set of generators and relations. See also * Elementary divisors References * Chap.8, p.128. * Chapter III.7, p.153 of Module theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chinese Remainder Theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). For example, if we know that the remainder of ''n'' divided by 3 is 2, the remainder of ''n'' divided by 5 is 3, and the remainder of ''n'' divided by 7 is 2, then without knowing the value of ''n'', we can determine that the remainder of ''n'' divided by 105 (the product of 3, 5, and 7) is 23. Importantly, this tells us that if ''n'' is a natural number less than 105, then 23 is the only possible value of ''n''. The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the '' Sun-tzu Suan-ching'' in the 3rd century CE. The Chinese remainder theorem is widely used for computing with lar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Factors
The invariant factors of a module (mathematics), module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R is a Principal ideal domain, PID and M a Finitely-generated module, finitely generated R-module, then :M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m) for some integer r\geq0 and a (possibly empty) list of nonzero elements a_1,\ldots,a_m\in R for which a_1 \mid a_2 \mid \cdots \mid a_m. The nonnegative integer r is called the ''free rank'' or ''Betti number'' of the module M, while a_1,\ldots,a_m are the ''invariant factors'' of M and are unique up to associatedness. The invariant factors of a Matrix (mathematics), matrix over a PID occur in the Smith normal form and provide a means of computing the structure of a module from a set of generators and relations. See also * Elementary divisors References * Chap.8, p.128. * Chapter III.7, p.153 of Module theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
