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In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a.
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space.
Nathan Jacobson was an American mathematician.
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for a ∈ R and s ∈ S, the intersection aS ∩ sR ≠ ∅. A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.
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 ; 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.
Irving Kaplansky was a mathematician, college professor, author, and amateur musician.
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. The semifir property is left-right symmetric, but the fir property is not.
Leonid Nisonovich Vaserstein is a Russian-American mathematician, currently Professor of Mathematics at Penn State University. His research is focused on algebra and dynamical systems. He is well known for providing a simple proof of the Quillen–Suslin theorem, a result in commutative algebra, first conjectured by Jean-Pierre Serre in 1955, and then proved by Daniel Quillen and Andrei Suslin in 1976.
Paul Moritz Cohn FRS was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings.
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 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.
In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring over which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book.
In abstract algebra, a uniserial moduleM is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts.
In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If N is a dense submodule of M, it may alternatively be said that "N ⊆ M is a rational extension". Dense submodules are connected with rings of quotients in noncommutative ring theory. Most of the results appearing here were first established in, and.
In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian group M which commutes with all R-endomorphisms of M is given by multiplication by a ring element. Explicitly, for any additive endomorphism f, if fg = gf for every R endomorphism g, then there exists an r in R such that f(x) = xr for all x in M. In the case of non-balanced modules, there will be such an f that is not expressible this way.
References
- Cohn, P. M. (2000), "From Hermite rings to Sylvester domains", Proceedings of the American Mathematical Society , 128 (7): 1899–1904, doi: 10.1090/S0002-9939-99-05189-8 , ISSN 0002-9939, MR 1646314
- Cohn, P. M. (2006), Free Ideal Rings and Localization in General Rings, Cambridge University Press, ISBN 9780521853378
- Kaplansky, Irving (1949), "Elementary divisors and modules", Transactions of the American Mathematical Society , 66 (2): 464–491, doi: 10.2307/1990591 , ISSN 0002-9947, JSTOR 1990591, MR 0031470
- Lam, T. Y. (1978), Serre's Conjecture, Lecture Notes in Mathematics, vol. 635, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068340, ISBN 978-3-540-08657-4, MR 0485842
- Lam, T. Y. (2006), Serre's Problem on Projective Modules, Springer Monographs in Mathematics, Berlin, Heidelberg: Springer-Verlag, doi:10.1007/978-3-540-34575-6, ISBN 978-3-540-23317-6
- Lissner, David (1965), "Outer product rings", Transactions of the American Mathematical Society , 116: 526–535, doi: 10.2307/1994132 , ISSN 0002-9947, JSTOR 1994132, MR 0186687
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