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Minimal Ideal
In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring ''R'' is a nonzero right ideal which contains no other nonzero right ideal. Likewise, a minimal left ideal is a nonzero left ideal of ''R'' containing no other nonzero left ideals of ''R'', and a minimal ideal of ''R'' is a nonzero ideal containing no other nonzero two-sided ideal of ''R'' . In other words, minimal right ideals are minimal elements of the poset of nonzero right ideals of ''R'' ordered by inclusion. The reader is cautioned that outside of this context, some posets of ideals may admit the zero ideal, and so the zero ideal could potentially be a minimal element in that poset. This is the case for the poset of prime ideals of a ring, which may include the zero ideal as a minimal prime ideal. Definition The definition of a minimal right ideal ''N'' of a ring ''R'' is equivalent to the following conditions: *''N'' is nonzero and if ''K'' is a right ideal of ''R'' with , then either ... [...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] |
Domain (ring Theory)
In algebra, a domain is a nonzero ring in which implies or .Lam (2001), p. 3 (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".Some authors also consider the zero ring to be a domain: see Polcino M. & Sehgal (2002), p. 65. Some authors apply the term "domain" also to rngs with the zero-product property; such authors consider ''n''Z to be a domain for each positive integer ''n'': see Lanski (2005), p. 343. But integral domains are always required to be nonzero and to have a 1. Examples and non-examples * The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0. More generally, for a positive integer ''n'', the ring Z/''n''Z is a domain if and only ... [...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] |
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 international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bimodules
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules. Definition If ''R'' and ''S'' are two rings, then an ''R''-''S''-bimodule is an abelian group (M,+) such that: # ''M'' is a left ''R''-module and a right ''S''-module. # For all ''r'' in ''R'', ''s'' in ''S'' and ''m'' in ''M'': (r.m).s = r.(m.s) . An ''R''-''R''-bimodule is also known as an ''R''-bimodule. Examples * For positive integers ''n'' and ''m'', the set ''M''''n'',''m''(R) of matrices of real numbers is an ''R''-''S''-bimodule, where ''R'' is the ring ''M''''n''(R) of matrices, and ''S'' is the ring ''M''''m''(R) of matrices. Addition and multiplication are carried out using the usual ru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiprime Ring
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced rings. For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form n\mathbb Z where ''n'' is a square-free integer. So, 30\mathbb Z is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but 12\mathbb Z\, is not (because 12 = 22 × 3, with a repeated prime factor). The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings. Most definitions and assertions in this article appear in and . Definitions For a commutative ring ''R'', a proper ideal ''A'' is a semiprime ideal if ''A'' satisfies either of the following equivalent conditions: * If ''x''''k'' is in ''A'' for some positive integer ''k'' and element ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see ''Semisimple representation''. Definition A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module ''M'', the following are equivalent: # ''M'' is semisimple; i.e. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Ring
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and ''simple ring'' are synonyms. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial ideals (since any ideal of M_n(R) is of the form M_n(I) with I an ideal of R), but has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns). A ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept. Definitions * a ''left principal ideal'' of R is a subset of R given by Ra = \ for some element a, * a ''right principal ideal'' of R is a subset of R given by aR = \ for some element a, * a ''two-sided principal ideal'' of R is a subset of R given by RaR = \ for some element a, namely, the set of all finite sums of elements of the form ras. While this definition for two-sided principal ideal may seem more complicated than the others, it is necessary to ensure that the ideal remains closed under addition. If R is a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
