- published: 30 Dec 2013
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In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.
The conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they also proved to be useful in other branches of mathematics such as geometry and mathematical analysis.
A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication.
Learn the definition of a ring, one of the central objects in abstract algebra. We give several examples to illustrate this concept including matrices and polynomials. Teaching assistant: Liliana de Castro Written & directed by Michael Harrison Produced by Kimberly Hatch Harrison
Building on the idea of groups, this lecture explores the structures called rings and fields, beginning to more closely resemble the number systems we work with every day.
Ring Theory: We define rings and give many examples. Items under consideration include commutativity and multiplicative inverses. Example include modular integers, square matrices, polynomial rings, quaternions, and adjoins of algebraic and transcendental numbers.
Introduction to rings: defining a ring and giving examples. Knowledge of sets, proofs, and mathematical groups are recommended. Practice problems: 1.) Which of the following sets are rings? If it is, prove it. If not, say which property of rings fails for that set (there may be more than one). a) N = {1,2,3,4,5,...} under normal addition and multiplication b) A = {a+b*sqrt(2) | a,b are rationals} under normal addition and multiplication c) B = {all polynomials p(x) with integer coefficients} d) C = {all polynomials p(x) with integer coefficients, where deg( p(x) ) is even} 2.) The set {0,2,4,6,8,10,12} is a ring with unity under the operations of addition mod 14 and multiplication mod 14. What is the unity of this ring? 3.) Let R be a commutative ring with unity, and let U(R) de...
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Annulus (Ring),8th Class Mathematics. English Medium. AP And Telangana State English Medium syllabus Online by Digital Teacher.
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Module (mathematics) In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module.Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=fOmgqXMTljI
Ring (mathematics) In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication.Through this generalization, theorems from arithmetic are extended to non-numerical objects like polynomials, series, matrices and functions. =======Image-Copyright-Info======= Image is in public domain Author-Info: David Hilbert Image Source: https://en.wikipedia.org/wiki/File:Chapitel_IX._of_Die_Theorie_der_algebraischen_Zahlkörper.png =======Image-Copyright-Info======== -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=FQoIuwlvvrc
Annulus (mathematics) In mathematics, an annulus (the Latin word for "little ring", with plural annuli) is a ring-shaped object, especially a region bounded by two concentric circles.The adjectival form is annular (as in annular eclipse). =======Image-Copyright-Info======== License: Creative Commons Attribution-Share Alike 3.0 (CC BY-SA 3.0) LicenseLink: http://creativecommons.org/licenses/by-sa/3.0 Author-Info: Dmcq Image Source: https://en.wikipedia.org/wiki/File:Annulus_area.svg =======Image-Copyright-Info======== -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=5xN6Qtqf1mY
Principal ideal 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. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=grMLOqjUGpA
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Ring Theory: We define ideals in rings as an analogue of normal subgroups in group theory. We give a correspondence between (two-sided) ideals and kernels of homomorphisms using quotient rings. We also state the First Isomorphism Theorem for Rings and give examples.
In this video we discuss the construction of quotient rings and the first isomorphism theorem.
In this video we discuss the construction of quotient rings and the first isomorphism theorem.
In this video we define formal polynomials and construct the ring of polynomials over a ring R.
Visual Group Theory, Lecture 7.2: Ideals, quotient rings, and finite fields A left (resp., right) ideal of a ring R is a subring that is invariant under left (resp., right) multiplication. Two-sided ideals are those that are both left and right ideals. This is the analogue of normal subgroups, in that the quotient ring R/I is well-defined iff I is a two-sided ideal. If one takes the quotient of the polynomial ring over Z_p (the integers modulo p) by an irreducible polynomial of degree n, then the result is a finite field of order q=p^n. It turns out that up to isomorphism, there is a unique finite field of each order q=p^n. Thus, all finite fields are of the form Z_p for a prime p, or Z_p[x]/(f) for an irreducible polynomial f. Course webpage (with lecture notes, HW, etc.): http://www.ma...
In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields. Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization. However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals. This interesting area of number theory does have some foun...
http://www.pensieve.net/course/13 In this video, I give definitions, examples, and proofs talking about Rings. Hope you like it. Feel free to give me some constructive feedback! Works Cited: The only significant source for these videos is 'A First Course in Abstract Algebra', 2nd ed. by John Fraleigh. Though most of the problems and definitions come from this book, I explain the all the material & solutions in my own words.
In this video we introduce the definition of a mathematical ring.