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 and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.
Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.
There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively. The multiplication symbol ⋅ is omitted, so the juxtaposition of ring elements is interpreted as multiplication.
For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplication does not have to have an inverse. A commutative ring such; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique. For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elements. Equip the set Z 4 = with the following operat
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.. It is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t 1 / 2 with integer coefficients. Suppose we have an oriented link L, given as a knot diagram. We will define the Jones polynomial, V, using Kauffman's bracket polynomial, which we denote by ⟨ ⟩. Note that here the bracket polynomial is a Laurent polynomial in the variable A with integer coefficients. First, we define the auxiliary polynomial X = − w ⟨ L ⟩, where w denotes the writhe of L in its given diagram; the writhe of a diagram is the number of positive crossings minus the number of negative crossings. The writhe is not a knot invariant. X is a knot invariant since it is invariant under changes of the diagram of L by the three Reidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves.
The bracket polynomial is known to change by multiplication by − A ± 3 under a type I Reidemeister move. The definition of the X polynomial given above is designed to nullify this change, since the writhe changes appropriately by + 1 or − 1 under type I moves. Now make the substitution A = t − 1 / 4 in X to get the Jones polynomial V; this results in a Laurent polynomial with integer coefficients in the variable t 1 / 2. This construction of the Jones polynomial for tangles is a simple generalization of the Kauffman bracket of a link; the construction was developed by Vladimir Turaev and published in 1990. Let k be a non-negative integer and S k denote the set of all isotopic types of tangle diagrams, with 2 k ends, having no crossing points and no closed components. Turaev's construction makes use of the previous construction for the Kauffman bracket and associates to each 2 k -end oriented tangle an element of the free R -module R, where R is the ring of Laurent polynomials with integer coefficients in the variable t 1 / 2.
Jones' original formulation of his polynomial came from his study of operator algebras. In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which arose while studying certain models, e.g. the Potts model, in statistical mechanics. Let a link L be given. A theorem of Alexander's states that it is the trace closure of a braid, say with n strands. Now define a representation ρ of the braid group on n strands, Bn, into the Temperley–Lieb algebra TL n with coefficients in Z and δ = − A 2 − A − 2; the standard braid generator σ i is sent to A ⋅ e i + A − 1 ⋅ 1, where 1, e 1, …, e n − 1 are the standard generators of the Temperley–Lieb algebra. It can be checked that this defines a representation. Take the braid word σ obtained from L and compute δ n − 1 tr ρ where tr is the Markov trace; this gives ⟨ L ⟩
Pierre Alphonse Laurent
Pierre Alphonse Laurent was a French mathematician and Military Officer best known as the discoverer of the Laurent series, an expansion of a function into an infinite power series, generalizing the Taylor series expansion. He was born in France. Pierre Laurent entered the École Polytechnique in Paris in 1830, Laurent graduated from the École Polytechnique in 1832, being one of the best students in his year, entered the engineering corps as second lieutenant, he attended the École d'Application at Metz until he was sent to Algeria. Laurent returned to France from Algeria around 1840 and spent six years directing operations for the enlargement of the port of Le Havre on the English Channel coast. Rouen had been the main French port up to the nineteenth century but the hydraulic construction projects on which Laurent worked in Le Havre turned it into France's main seaport, it is clear that Laurent was a good engineer, putting his deep theoretical knowledge to good practical use. It was while Laurent was working on the construction project at Le Havre that he began to write his first mathematical papers.
He submitted a memoir for the Grand Prize of the Académie des Sciences of 1842. His result was contained in a memoir submitted for the Grand Prize of the Académie des Sciences in 1843, but his submission was after the due date, the paper was not published and never considered for the prize. Laurent died at age 41 in Paris, his work was not published until after his death. Laurent polynomial O'Connor, John J.. "Pierre Alphonse Laurent", MacTutor History of Mathematics archive, University of St Andrews
Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation, associative and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, such that every element's expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis. Addition of integers is commutative and has subtraction as its inverse operation, each integer is the sum or difference of some number of copies of the number 1, each integer has a unique representation as an integer multiple of the number 1. Free abelian groups have properties, they have applications in algebraic topology, where they are used to define chain groups, in algebraic geometry, where they are used to define divisors. Integer lattices form examples of free abelian groups, lattice theory studies free abelian subgroups of real vector spaces.
The elements of a free abelian group with basis B may be described in several equivalent ways. These include formal sums over B, which are expressions of the form ∑ a i b i where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; every set B has a free abelian group with B as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B.
Alternatively, the free abelian group with basis B may be described by a presentation with the elements of B as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every subgroup of a free abelian group is itself free abelian. The integers, under the addition operation, form a free abelian group with the basis; every integer n is a linear combination of basis elements with integer coefficients: namely, n = n × 1, with the coefficient n. The two-dimensional integer lattice, consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis. Letting these basis vectors be denoted e 1 = and e 2 =, the element can be written = 4 e 1 + 3 e 2 where'multiplication' is defined so that 4 e 1:= e 1 + e 1 + e 1 + e 1. In this basis, there is no other way to write. However, with a different basis such as, where f 1 = and f 2 =, it can be written as = f 1 + 3 f 2.
More every lattice forms a finitely-generated free abelian group. The d-dimensional integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if M is a d × d integer matrix with determinant ±1 the rows of M form a basis, conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods; the direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. More the direct product of any finite number of free abelian groups is free abelian; the d-dimensional integer lattice, for instance, is isomorphic to the direct product of d copies of the integer group Z. The trivial group is considered to be free abelian, with basis the empty set, it may be interpreted as a direct product of zero copies of Z. For infinite families of free abelian groups, the direct product is not free abelian. For instance the Baer–Specker group Z N