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Hermite Ring
In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to (p. 465), a ring is right Hermite if, for every two elements ''a'' and ''b'' of the ring, there is an element ''d'' of the ring and an invertible 2 by 2 matrix ''M'' over the ring such that ''(a b)M=(d 0)''. (The term left Hermite is defined similarly.) Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix (, p. 468.) (appendix to §I.4) calls this property K-Hermite, using ''Hermite'' instead in the sense given below. According to (§I.4, p. 26), a ring is right Hermite if any finitely generated stably free right module over the ring is free. This is equivalent to requiring that any row vector ''(b1,...,bn)'' of elements of the ring which generate it as a right module (i.e., ''b1R+...+bnR=R'') can be completed to a (not necessarily square) invertible matrix by adding some number of rows. (The criteri ... [...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 groups, rings, fields, modules, vector spaces, lattices, and 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 variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form 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 example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré. He was the first to prove that '' e'', the base of natural logarithms, is a transcendental number. His methods were used later by Ferdinand von Lindemann to prove that π is transcendental. Life Hermite was born in Dieuze, Moselle, on 24 December 1822, with a deformity in his right foot that would impair his gait throughout his life. He was the sixth of seven children of Ferdinand Hermite and his wife, Madeleine née Lallemand. Ferdinand worked in the drapery business of Madeleine's family while also pursuing a career as an artist. The drapery business relocate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Normal Form
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only. Other applications of the Hermite normal form include integer programming, cryptography, and abstract algebra. Definition Various authors may prefer to talk about Hermite normal form in either row-style or column-style. They are essentially the same up to transposition. Row-style Hermite normal form An m by n matrix A with integer entries has a (row) Hermite normal form H if there is a square unimodular matrix U where H=UA and H has the following restrictions: # H is upper triangular (that is, h''ij'' = 0 for ''i'' > ''j''), and any rows of zeros are located below any other row. # The leading c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stably Free Module
In mathematics, a stably free module is a module which is close to being free. Definition A finitely generated module ''M'' over a ring ''R'' is ''stably free'' if there exist free finitely generated modules ''F'' and ''G'' over ''R'' such that : M \oplus F = G . \, Properties * A projective module is stably free if and only if it possesses a finite free resolution. * An infinitely generated module is stably free if and only if it is free. See also * Free object * Eilenberg–Mazur swindle * Hermite ring In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to (p. 465), a ring is right Hermite if, for every two elements ''a'' and ''b'' of the ring, there is an element ''d'' of the ring and ... References {{Reflist Module theory Free algebraic structures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Basis Number
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over ''R'' have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension. Definition A ring ''R'' has invariant basis number (IBN) if for all positive integers ''m'' and ''n'', ''R''''m'' isomorphic to ''R''''n'' (as left ''R''-modules) implies that . Equivalently, this means there do not exist distinct positive integers ''m'' and ''n'' such that ''R''''m'' is isomorphic to ''R''''n''. Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever ''A'' is an ''m''-by-''n'' matrix over ''R'' and ''B'' is an ''n''-by-''m'' matrix over ''R'' such that and , then . This form reveals that the definition is left–right symmetric, so it makes no difference whether we define IBN in terms of left or right mo ... [...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, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout Ring
In mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when ''R'' is a commutative ring, ''R'' can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of ''R'' are principal, then ''R'' is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. A commutative principal ideal ring which is also an integral domain is said to be a ''principal ideal domain'' (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain. General properties If ''R'' is a principal right ideal ring, then it is certainly a right Noetherian rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
