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Projective Line Over A Ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring ''A'' with 1, the projective line P(''A'') over ''A'' consists of points identified by projective coordinates. Let ''U'' be the group of units of ''A''; pairs (''a, b'') and (''c, d'') from are related when there is a ''u'' in ''U'' such that and . This relation is an equivalence relation. A typical equivalence class is written ''U'' 'a, b'' that is, ''U'' 'a, b''is in the projective line if the ideal generated by ''a'' and ''b'' is all of ''A''. The projective line P(''A'') is equipped with a group of homographies. The homographies are expressed through use of the matrix ring over ''A'' and its group of units ''V'' as follows: If ''c'' is in Z(''U''), the center of ''U'', then the group action of matrix \beginc & 0 \\ 0 & c \end on P(''A'') is the same as the action of the identity matrix. Such matrices represent a normal subgroup ''N'' of ''V''. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projectivisation F7P%5E1
In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional Linear subspace, subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multiplication defines a subset of (V) formed by the lines contained in ''S'' and is called the projectivization of ''S''. Properties * Projectivization is a special case of the Quotient space (topology), factorization by a Group action (mathematics), group action: the projective space (V) is the quotient of the open set ''V''\ of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension (mathematics), dimension of (V) in the sense of algebraic geometry is one less than the dimension of the vector space ''V''. * Projectivization is functorial with respect to injective linear maps: if :: f: V\to W : is a linear map with trivial kernel (linear algebra), kernel then ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GF(2)
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with the notation of -adic integers. is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its logical foundations. Definition GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below: If the elements of GF(2) are seen as boolean values, then the addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout's Identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that , x, \le , b/d , and , y, \le , a/d , ; equality occurs only if one of and is a multiple of the other. As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that resu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a local ring, and the maximal right ideal is also the unique maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Harmonic Conjugate
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line through meet at respectively. If and meet at , and meets at , then is called the harmonic conjugate of with respect to . The point does not depend on what point is taken initially, nor upon what line through is used to find and . This fact follows from Desargues theorem. In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as . Cross-ratio criterion The four points are sometimes called a harmonic range (on the real projective line) as it is found that always divides the segment ''internally'' in the same proportion as divides ''externally''. That is: :, AC, :, BC, = , AD, :, DB, \, . If these segments are now endowed with the ordinary metric interpretation of real num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Geometry
Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Galois geometry may be defined as a projective space over a finite field. Objects of study include affine and projective spaces over finite fields and various structures that are contained in them. In particular, arcs, ovals, hyperovals, unitals, blocking sets, ovoids, caps, spreads and all finite analogues of structures found in non-finite geometries. Vector spaces defined over finite fields play a significant role, especially in construction methods. Projective spaces over finite fields Notation Although the generic notation of projective geometry is sometimes used, it is more common to denote projective spaces over finite fields by , where is the "geometric" dimension (see below), and is the order of the finite field (or Galois ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit (ring Theory)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a 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 sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More generally, any root of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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