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Invertible Element
In the branch of abstract algebra known as ring theory, a unit of a ring (mathematics), ring R is any element u \in R that has a multiplicative inverse in R: an element v \in R such that :vu = uv = 1, where is the multiplicative identity. The set of units of forms a Group (mathematics), group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is also used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''Ring (mathematics)#Multiplicative identity: mandatory vs. optional, unit ring'', and also e.g. ''identity matrix, 'unit' matrix''. For this reason, some authors call "unity" or "identity", and say that is a "ring with unity" or a "ring with identity" rather than a "ring with a unit". Examples The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring is a unit ...
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Ring Theory
In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ..., ring theory is the study of rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...—algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...s in which addition and multiplication are defined and have similar properties to those operations defined for the integer An integer (from the Latin Latin ...
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Coprime
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., two integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...s and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... of both of them is 1. Consequently, any prime number A prime number (or a prime) is a natural numbe ...
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General Linear Group
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., the general linear group of degree ''n'' is the set of invertible matrices In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ..., together with the operation of ordinary matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu .... This forms a group A group is a number A number is a mathematical object ...
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Square Matrix
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a square matrix is a matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ... with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
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Power Series Ring
Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power Power may also refer to: Mathematics, science and technology Computing * IBM POWER (software), an IBM operating system enhancement package * IBM POWER instruction set architecture, a RISC instruction set architecture * Power ISA, a RISC instruction set architecture derived from PowerPC * IBM Power microprocessors, made by IBM, which implement those RISC architectures * Power.org, a predecessor to the OpenPOWER Foundation * SGI POWER Challenge, a line of SGI supercomputers Mathematics * Exponentiation, "''x'' to the power of ''y''" * Power function * Power of a point * Statistical power Physics * Magnification, the factor by which an optical system enlarges an image * Optical power, the degree to which a lens converges or diverges light Social sciences and politics * Economic power, encompassing several concepts that economists use, feat ...
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Domain (ring Theory)
In algebra, an area of mathematics, a domain is a zero ring, nonzero ring (mathematics), ring in which implies or .Lam (2001), p. 3 (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative ring, commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".Some authors also consider the zero ring to be a domain: see Polcino M. & Sehgal (2002), p. 65. Some authors apply the term "domain" also to rng (mathematics), rngs with the zero-product property; such authors consider ''n''Z to be a domain for each positive integer ''n'': see Lanski (2005), p. 343. But integral domains are always required to be nonzero and to have a 1. Examples and non-examples * The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0. ...
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Nilpotent
In mathematics, an element ''x'' of a ring (mathematics), ring ''R'' is called nilpotent if there exists some positive integer ''n'', called the index (or sometimes the degree), such that ''x''''n'' = 0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrix, square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because ''A''3 = 0. See nilpotent matrix for more. * In the factor ring Z/9Z, the equivalence class of 3 is nilpotent because 32 is Congruence relation, congruent to 0 Modular arithmetic, modulo 9. * Assume that two elements ''a'', ''b'' in a ring ''R'' satisfy ''ab'' = 0. Then the element ''c'' = ''ba'' is nilpotent as ''c''2 = (''ba'')2 = ''b''(''ab'')''a'' = 0. An example with matrices (for ''a'', ''b''): :: A = \be ...
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Polynomial Ring
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., especially in the field of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ..., a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...) formed from the set of polynomial In mathematics Mathematics (from Ancient Greek, ...
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Real Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over , the rational numbers. Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and for an imaginary quadratic field or complex quadratic field, corresponding to whether or not it is a Field extension, subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important. Ring of integers Discriminant For a nonzero square free integer ''d'', the Discriminant of an algebraic number field, discriminant of the quadratic field ''K''=Q() is ''d'' if ''d'' is congruent to 1 modulo 4, and otherwise 4''d''. For example, if ''d'' is −1, then ''K'' is th ...
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Rank Of A Module
In mathematics, a finitely generated module is a module (mathematics), module that has a Finite set, finite generating set. A finitely generated module over a Ring (mathematics), ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a Field (mathematics), field is simply a Dimension (vector space), finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. Definition The left ''R''-module ''M'' is finitely generated if there exist ''a''1, ''a''2, ..., ''a''''n'' in ''M'' such that for any ''x'' in ''M'', there exist ''r''1, ''r''2, ..., ''r''''n'' in ''R'' with ''x'' ...
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Dirichlet's Unit Theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of an abelian group, rank of the group of units in the ring (mathematics), ring of algebraic integers of a number field . The regulator is a positive real number that determines how "dense" the units are. The statement is that the group of units is finitely generated and has Rank of an abelian group, rank (maximal number of multiplicatively independent elements) equal to : where is the ''number of real embeddings'' and the ''number of conjugate pairs of complex embeddings'' of . This characterisation of and is based on the idea that there will be as many ways to embed in the complex number field as the degree n = [K: \mathbb]; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that :. Note that if is Galois over \mathbb then either or . Other ways of determining and are * ...
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Number Field
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an algebraic number field (or simply number field) K is a finite degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ... (and hence algebraic) field extension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... of the field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for a ...
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