1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Ring (mathematics)
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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 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, 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 proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −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 1. R is a 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.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity

3.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

4.
Zero object (algebra)
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In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, the aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as. One often refers to the trivial object since every trivial object is isomorphic to any other, instances of the zero object include, but are not limited to the following, As a group, the trivial group. As a ring, the trivial ring, as an algebra over a field or algebra over a ring, the trivial algebra. As a module, the zero module, the term trivial module is also used, although it may ambiguous, as a trivial G-module is a G-module with a trivial action. As a vector space, the vector space, zero-dimensional vector space or just zero space. These objects are described not only based on the common singleton and trivial group structure. In the last three cases the scalar multiplication by an element of the ring is defined as, κ0 = 0 . The most general of them, the module, is a finitely-generated module with an empty generating set. For structures requiring the structure inside the zero object, such as the trivial ring, there is only one possible,0 ×0 =0. This structure is associative and commutative, a ring R which has both an additive and multiplicative identity is trivial if and only if 1 =0, since this equality implies that for all r within R, r = r ×1 = r ×0 =0. In this case it is possible to define division by zero, some properties of depend on exact definition of the multiplicative identity, see the section Unital structures below. Any trivial algebra is also a trivial ring, a trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, an algebra is simultaneously a zero module. The trivial ring is an example of a rng of square zero, a trivial algebra is an example of a zero algebra. The zero-dimensional vector space is a ubiquitous example of a zero object. It is also a group over addition, and a trivial module mentioned above. The trivial ring, zero module and zero vector space are zero objects of the categories, namely Rng, R-Mod

5.
Boolean ring
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In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, such as the ring of integers modulo 2. That is, R consists only of idempotent elements, every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference. Boolean rings are named after the founder of Boolean algebra, George Boole, there are at least four different and incompatible systems of notation for Boolean rings and algebras. In commutative algebra the standard notation is to use x + y = ∨ for the sum of x and y. In logic, a notation is to use x ∧ y for the meet and use x ∨ y for the join. In set theory and logic it is common to use x · y for the meet. This use of + is different from the use in ring theory, a rare convention is to use xy for the product and x ⊕ y for the ring sum, in an effort to avoid the ambiguity of +. Historically, the term Boolean ring has been used to mean a Boolean ring possibly without an identity, one example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can consider the set of all finite or cofinite subsets of X, again with symmetric difference. More generally with these operations any field of sets is a Boolean ring, by Stones representation theorem every Boolean ring is isomorphic to a field of sets. Since the join operation ∨ in a Boolean algebra is written additively, it makes sense in this context to denote ring addition by ⊕. Given a Boolean ring R, for x and y in R we can define x ∧ y = xy, x ∨ y = x ⊕ y ⊕ xy and these operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring becomes a Boolean algebra, similarly, every Boolean algebra becomes a Boolean ring thus, xy = x ∧ y, x ⊕ y = ∧ ¬. If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the analogous result holds beginning with a Boolean algebra. A map between two Boolean rings is a homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is an ideal if. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the algebra of the corresponding Boolean algebra modulo the corresponding order ideal. The property x ⊕ x =0 shows that any Boolean ring is an algebra over the field F2 with two elements, in just one way

6.
Clifford algebra
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In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions, the theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and they are named after the English geometer William Kingdon Clifford. The most familiar Clifford algebra, or orthogonal Clifford algebra, is referred to as Riemannian Clifford algebra. A Clifford algebra is an associative algebra that contains and is generated by a vector space V over a field K. One common way of writing this is to say that the algebra generated by V may be written as the tensor algebra ⊕n≥0 V ⊗. The product induced by the product in the quotient algebra is written using juxtaposition. Its associativity follows from the associativity of the tensor product, the definition of a Clifford algebra endows the algebra with more structure than a bare K-algebra, specifically it has a distinguished subspace V. Such a subspace cannot in general be uniquely determined only a K-algebra isomorphic to the Clifford algebra. The idea of being the freest or most general algebra subject to identity can be formally expressed through the notion of a universal property. Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case, in particular, if char =2 it is not true that a quadratic form uniquely determines a symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this include the condition that the characteristic is not 2. Clifford algebras are related to exterior algebras. In fact, if Q =0 then the Clifford algebra Cℓ is just the exterior algebra Λ, for nonzero Q there exists a canonical linear isomorphism between Λ and Cℓ whenever the ground field K does not have characteristic two. That is, they are isomorphic as vector spaces. Clifford multiplication together with the subspace is strictly richer than the exterior product since it makes use of the extra information provided by Q. A different way of saying this is that, if one takes the Clifford algebra to be a filtered algebra, then the associated graded algebra is the exterior algebra. More precisely, Clifford algebras may be thought of as quantizations of the exterior algebra, Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras

7.
Commutative ring
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In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not or is not required to be commutative. e, operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by + and ⋅, e. g. a + b, the identity elements for addition and multiplication are denoted 0 and 1, respectively. If the multiplication is commutative, i. e. a ⋅ b = b ⋅ a, in the remainder of this article, all rings will be commutative, unless explicitly stated otherwise. An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication, as the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen, a field is a commutative ring where every non-zero element a is invertible, i. e. has a multiplicative inverse b such that a ⋅ b =1. Therefore, by definition, any field is a commutative ring, the rational, real and complex numbers form fields. An example is the set of matrices of divided differences with respect to a set of nodes. If R is a commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring. The same holds true for several variables, if V is some topological space, for example a subset of some Rn, real- or complex-valued continuous functions on V form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold, in contrast to fields, where every nonzero element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation, first, an element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i. e. a non-zero element a such that there exists an element b of the ring such that ab =0. If R possesses no zero divisors, it is called an integral domain since it resembles the integers in some ways. Many of the following notions also exist for not necessarily commutative rings, for example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably. Given any subset F = j ∈ J of R, the ideal generated by F is the smallest ideal that contains F. Equivalently, an ideal generated by one element is called a principal ideal. A ring all of whose ideals are principal is called a principal ideal ring, any ring has two ideals, namely the zero ideal and R, the whole ring