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searching for Unit (ring theory) 88 found (142 total)

Ideal (ring theory) (6,231 words) [view diff] no match in snippet view article

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the

Local ring (2,300 words) [view diff] no match in snippet view article find links to article

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local

Quotient ring (2,983 words) [view diff] no match in snippet view article find links to article

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite

Domain (ring theory) (914 words) [view diff] no match in snippet view article

irreducible components. Zero divisor Zero-product property Divisor (ring theory) Integral domain Lam (2001), p. 3 Rowen (1994), p. 99. Some authors also

Algebra over a field (3,122 words) [view diff] no match in snippet view article find links to article

doi:10.1007/BF01692479, S2CID 121426669 Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid

Square-free element (181 words) [view diff] no match in snippet view article find links to article

This means that every s such that s 2 ∣ r {\displaystyle s^{2}\mid r} is a unit of R. Square-free elements may be also characterized using their prime decomposition

Free algebra (915 words) [view diff] no match in snippet view article find links to article

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since

Topological ring (1,100 words) [view diff] no match in snippet view article find links to article

as ( x , x − 1 ) . {\displaystyle \left(x,x^{-1}\right).} However, if the unit group is endowed with the subspace topology as a subspace of R , {\displaystyle

Subring (895 words) [view diff] no match in snippet view article find links to article

{\displaystyle \mathbb {Z} \cup \{i\}} , and thus is the adjunction of the imaginary unit i to Z {\displaystyle \mathbb {Z} } . The intersection of all subrings of

Von Neumann regular ring (1,299 words) [view diff] no match in snippet view article find links to article

include unit regular rings and strongly von Neumann regular rings and rank rings. A ring R is called unit regular if for every a in R, there is a unit u in

Division algebra (1,153 words) [view diff] no match in snippet view article find links to article

non-associative division algebra of dimension 2 over the reals, and has no unit element. There are infinitely many other non-isomorphic commutative, non-associative

Nil ideal (734 words) [view diff] no match in snippet view article find links to article

In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if all of its elements is nilpotent

Semifield (770 words) [view diff] no match in snippet view article find links to article

as is usually found in definitions of fields and division rings. In ring theory, combinatorics, functional analysis, and theoretical computer science

Cartan–Brauer–Hua theorem (138 words) [view diff] no match in snippet view article find links to article

the center of D, or K = D. In other words, if the unit group of K is a normal subgroup of the unit group of D, then either K = D or K is central (Lam

Composition ring (905 words) [view diff] no match in snippet view article find links to article

series are absent, this composition ring does not have a multiplicative unit. If R is an integral domain, the field R(X) of rational functions also has

Semiring (8,021 words) [view diff] no match in snippet view article find links to article

y+(x\,m+y\,n),n\cdot m\rangle } defines another semiring with multiplicative unit 1 R × N := ⟨ 0 R , 1 N ⟩ {\displaystyle 1_{R\times {\mathbb {N} }}:=\langle

Commutative ring (5,688 words) [view diff] no match in snippet view article find links to article

ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that

Category of rings (1,814 words) [view diff] no match in snippet view article find links to article

product of rings ⊗Z as the monoidal product and the ring of integers Z as the unit object. It follows from the Eckmann–Hilton theorem, that a monoid in Ring

Fractional ideal (1,611 words) [view diff] no match in snippet view article find links to article

in Glaz, Sarah; Chapman, Scott T. (eds.), Non-Noetherian commutative ring theory, Mathematics and its Applications, vol. 520, Dordrecht: Kluwer Acad.

Idempotent (ring theory) (2,327 words) [view diff] no match in snippet view article

In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is

Classification of Clifford algebras (2,420 words) [view diff] no match in snippet view article find links to article

quadratic form, the unit pseudoscalar is a volume form), and lifts reflection through the origin (meaning that the image of the unit pseudoscalar is reflection

Dyadic rational (3,895 words) [view diff] no match in snippet view article find links to article

halving, which produces dyadic rationals when measuring fractional amounts of units. The inch is customarily subdivided in dyadic rationals rather than using

Jacobson radical (2,891 words) [view diff] no match in snippet view article find links to article

In mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle

Dyadic rational (3,895 words) [view diff] no match in snippet view article find links to article

halving, which produces dyadic rationals when measuring fractional amounts of units. The inch is customarily subdivided in dyadic rationals rather than using

Solenoid (mathematics) (1,488 words) [view diff] no match in snippet view article

defined mod 2π) and consider the complex coordinate z on the two-dimensional unit disk D. Let f be the map of the solid torus T = S1 × D into itself given

Graded ring (2,820 words) [view diff] no match in snippet view article find links to article

Graded rings, the Hilbert function and the Samuel function". Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid

Matrix ring (1,812 words) [view diff] no match in snippet view article find links to article

C*-algebra, then Mn(A) is another C*-algebra. If A is non-unital, then Mn(A) is also non-unital. By the Gelfand–Naimark theorem, there exists a Hilbert

Flat module (4,590 words) [view diff] no match in snippet view article find links to article

(1970), Commutative algebra Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University

Group ring (3,982 words) [view diff] no match in snippet view article find links to article

R-algebra to its group of units. When R = Z, this gives an adjunction between the category of groups and the category of rings, and the unit of the adjunction

Semi-local ring (446 words) [view diff] no match in snippet view article find links to article

Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m1, ..., mn R / ⋂ i = 1 n m i ≅ ⨁ i = 1 n R / m i {\displaystyle

Ideal norm (1,079 words) [view diff] no match in snippet view article find links to article

} since Z {\displaystyle \mathbb {Z} } has trivial ideal class group and unit group { ± 1 } {\displaystyle \{\pm 1\}} , thus each nonzero fractional ideal

Complex-base system (1,965 words) [view diff] no match in snippet view article find links to article

include the following ( i {\displaystyle \mathrm {i} } being the imaginary unit): ⟨ R , Z R ⟩ {\displaystyle \left\langle {\sqrt {R}},Z_{R}\right\rangle

Kaplansky's theorem on projective modules (1,886 words) [view diff] no match in snippet view article find links to article

"Bass's work in ring theory and projective modules". arXiv:math/0002217. MR1732042 Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies

Novikov ring (740 words) [view diff] no match in snippet view article find links to article

]} consisting of those with leading term 1. Since the elements of S are unit elements of Nov ⁡ ( Γ ) {\displaystyle \operatorname {Nov} (\Gamma )} , the

Zero-product property (1,261 words) [view diff] no match in snippet view article find links to article

functions f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } , from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs

Torsion (algebra) (1,660 words) [view diff] no match in snippet view article

In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of

Dedekind-finite ring (146 words) [view diff] no match in snippet view article find links to article

is Dedekind-finite. A unit-regular ring is Dedekind-finite. A local ring is Dedekind-finite. Goodearl, Kenneth (1976). Ring Theory: Nonsingular Rings and

Quantale (597 words) [view diff] no match in snippet view article find links to article

topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). Quantales

Multiple (mathematics) (593 words) [view diff] no match in snippet view article

submultiple of a foot, or a 36-fold submultiple of a yard. Unit fraction Ideal (ring theory) Decimal and SI prefix Multiplier (linguistics) Weisstein,

Associative algebra (4,261 words) [view diff] no match in snippet view article find links to article

unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative

Glossary of commutative algebra (9,772 words) [view diff] no match in snippet view article find links to article

classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed

Krull ring (2,692 words) [view diff] no match in snippet view article find links to article

Publishing Company Inc. ISBN 0-8053-7026-9. Hideyuki Matsumura, Commutative Ring Theory. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced

Non-associative algebra (3,005 words) [view diff] no match in snippet view article find links to article

Triple systems There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative

Crossed product (1,601 words) [view diff] no match in snippet view article find links to article

ring of a semidirect product group. Therefore crossed products have a ring theory aspect also. This article concentrates on an important case, where they

Profinite integer (2,133 words) [view diff] no match in snippet view article find links to article

group of Q Questions on some maps involving rings of finite adeles and their unit groups. Milne 2013, Ch. I Example A. 5. "Class field theory - lccs". www

Bialgebra (1,612 words) [view diff] no match in snippet view article find links to article

mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra.: 46 The algebraic

Opposite category (619 words) [view diff] no match in snippet view article find links to article

Clearly, the same construction works for groups, as well, and is known in ring theory, too, where it is applied to the multiplicative semigroup of the ring

Weyl algebra (4,164 words) [view diff] no match in snippet view article find links to article

the central element of the Heisenberg algebra (namely [q, p]) equal to the unit of the universal enveloping algebra (called 1 above). The algebra W(V) is

Prüfer group (1,137 words) [view diff] no match in snippet view article find links to article

Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product

Grade (581 words) [view diff] no match in snippet view article find links to article

(timber) quality and strength Grade (angle), a unit for the measurement of plane angles Grade (ring theory), a cohomological invariant in commutative algebra

Universal enveloping algebra (8,954 words) [view diff] no match in snippet view article find links to article

In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations

Algebraic variety (5,761 words) [view diff] no match in snippet view article find links to article

correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry. Many

Split-biquaternion (1,095 words) [view diff] no match in snippet view article find links to article

space of split-biquaternions forms a free module. This standard term of ring theory expresses a similarity to a vector space, and this structure by Clifford

Composition algebra (1,319 words) [view diff] no match in snippet view article find links to article

vector, N is an isotropic quadratic form, and "the algebra splits". Every unital composition algebra over a field K can be obtained by repeated application

List of theorems (6,289 words) [view diff] no match in snippet view article find links to article

Brauer–Cartan–Hua theorem (ring theory) Frobenius theorem (abstract algebras) Goldie's theorem (ring theory) Jacobson density theorem (ring theory) Jacobson–Bourbaki

Adjoint functors (10,260 words) [view diff] no match in snippet view article find links to article

via a method that is formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that might not have a multiplicative

Dual quaternion (4,786 words) [view diff] no match in snippet view article find links to article

subspace is called an ideal in ring theory. It happens to be the unique maximal ideal of the ring of dual numbers. The group of units of the dual number ring

Free module (1,808 words) [view diff] no match in snippet view article find links to article

Encyclopedia of Mathematics, EMS Press. Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University

Epimorphism (2,355 words) [view diff] no match in snippet view article find links to article

\varepsilon } ). There is also the notion of homological epimorphism in ring theory. A morphism f: A → B of rings is a homological epimorphism if it is an

Monoid (4,462 words) [view diff] no match in snippet view article find links to article

preordering ≤, defined by x ≤ y if there exists z such that x + z = y. An order-unit of a commutative monoid M is an element u of M such that for any element

Institute of Mathematics, UFRJ (278 words) [view diff] no match in snippet view article find links to article

partial differential equations Mathematical physics Differential geometry Ring theory and group theory System dynamics Geometric theory of foliations Algebraic

Octonion (5,316 words) [view diff] no match in snippet view article find links to article

Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly seven maximal orders containing it. These seven maximal

Almost all (2,577 words) [view diff] no match in snippet view article find links to article

a Subset?". In Hazewinkel, Michiel (ed.). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. Vol. 520. Springer. p. 85. doi:10

List of unsolved problems in mathematics (20,069 words) [view diff] no match in snippet view article find links to article

тетрадь) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory. The Erlagol Notebook (Russian: Эрлагольская тетрадь)

Emmy Noether (15,266 words) [view diff] no match in snippet view article find links to article

(in particular Levitzky's theorem and the Hopkins–Levitzki theorem) to ring theory. Other Noether Boys included Max Deuring, Hans Fitting, Ernst Witt, Chiungtze

Macsyma (2,534 words) [view diff] no match in snippet view article find links to article

Spear, David (July 1977). "A constructive approach to commutative ring theory". Proceedings of the 1977 MACSYMA Users' Conference. Mora, Teo (2005)

O-ring (3,361 words) [view diff] no match in snippet view article find links to article

rings (back-up rings) Cooper Ring Diaphragm seal Gasket Labyrinth seal O-ring theory of economic development Ozone cracking Polymer degradation Obturating

Algebraic K-theory (10,647 words) [view diff] no match in snippet view article find links to article

subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are

Vector space (11,491 words) [view diff] no match in snippet view article find links to article

written in the form x + iy for real numbers x and y where i is the imaginary unit, form a vector space over the reals with the usual addition and multiplication:

Projective module (3,081 words) [view diff] no match in snippet view article find links to article

Univ. Press. ISBN 0-691-08238-3. Donald S. Passman (2004) A Course in Ring Theory, especially chapter 2 Projective modules, pp 13–22, AMS Chelsea, ISBN 0-8218-3680-3

Boolean algebra (structure) (3,372 words) [view diff] no match in snippet view article

Minimal axioms for Boolean algebra. Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras"

Benzene (9,595 words) [view diff] no match in snippet view article find links to article

Wilcox, David H.; Greenbaum, Frederick R. (1965). "Kekule's benzene ring theory: A subject for lighthearted banter". Journal of Chemical Education. 42

Dedekind domain (3,715 words) [view diff] no match in snippet view article find links to article

R {\displaystyle yR} are equal iff x y − 1 {\displaystyle xy^{-1}} is a unit in R. For a general domain R, it is meaningful to take the quotient of the

Convexity in economics (3,070 words) [view diff] no match in snippet view article find links to article

convex if, for all points v0 and v1 in Q and for every real number λ in the unit interval [0,1], the point (1 − λ) v0 + λv1 is a member of Q. By mathematical

Finite field (7,535 words) [view diff] no match in snippet view article find links to article

{\displaystyle \mathrm {GF} (p)} consists of evenly spaced points around the unit circle (omitting zero). The field G F ( 64 ) {\displaystyle \mathrm {GF}

Carl Friedrich Gauss (17,930 words) [view diff] no match in snippet view article find links to article

Kleiner, Israel (1998). "From Numbers to Rings: The Early History of Ring Theory". Elemente der Mathematik. 53 (1): 18–35. doi:10.1007/s000170050029.

Special classes of semigroups (428 words) [view diff] no match in snippet view article find links to article

India, 1986 [Kela] A. V. Kelarev, Applications of epigroups to graded ring theory, Semigroup Forum, Volume 50, Number 1 (1995), 327-350 doi:10.1007/BF02573530

Shapley–Folkman lemma (10,580 words) [view diff] no match in snippet view article find links to article

set as its endpoints. The convex hull of the unit circle is the closed unit disk, which contains the unit circle and its interior. In any vector space

Euclidean algorithm (15,349 words) [view diff] no match in snippet view article find links to article

definition of the GCD is helpful in advanced mathematics, particularly ring theory. The greatest common divisor g of two nonzero numbers a and b is also

John von Neumann (23,699 words) [view diff] no match in snippet view article find links to article

traditional projective geometry with modern algebra (linear algebra, ring theory, lattice theory). Many previously geometric results could then be interpreted

Normal polytope (1,267 words) [view diff] no match in snippet view article find links to article

and d {\displaystyle d} . Convex cone Algebraic geometry Number theory Ring theory Ehrhart polynomial Rational cone Toric variety Stanley, Richard P. (1986)

Ohio University (9,465 words) [view diff] no match in snippet view article find links to article

Cartographic Center, the Institute for Quantitative Biology, and the Center for Ring Theory and Its Applications. The Center for International Studies was established

Gröbner basis (10,037 words) [view diff] no match in snippet view article find links to article

Mathematical Society. ISBN 0-8218-3804-0. Li, Huishi (2011). Gröbner Bases in Ring Theory. World Scientific. ISBN 978-981-4365-13-0. Becker, Thomas; Weispfenning

Charles I. Jones (2,243 words) [view diff] no match in snippet view article find links to article

“Intermediate Goods and Weak Links”. Beginning with Michael Kremer’s “The O-Ring Theory of Economic Development”, economists have recognized that production

List of people who disappeared mysteriously: 1910–1990 (10,921 words) [view diff] no match in snippet view article find links to article

June 2011. Retrieved 12 August 2012. "'Missing five sold off by HK vice ring' theory". The Straits Times. 20 September 1978. Retrieved 28 November 2023. "Kim

Neal Henry McCoy (1,569 words) [view diff] no match in snippet view article find links to article

McCoy, Neal H. (1950). "Some theorems on groups with applications to ring theory". Transactions of the American Mathematical Society. 69: 302–311. doi:10

Timeline of category theory and related mathematics (273 words) [view diff] no match in snippet view article find links to article

Jean-Pierre Serre Algebraic K-theory launched by explicit analogy of ring theory with geometric cases. 1960 Alexander Grothendieck Fiber functors 1960

History of mathematical notation (11,251 words) [view diff] no match in snippet view article find links to article

Klein develop the Kaluza–Klein theory. In 1928, Emil Artin abstracted ring theory with Artinian rings. In 1933, Andrey Kolmogorov introduces the Kolmogorov