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Primary Ideal
In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n'' > 0. For example, in the ring of integers Z, (''p''''n'') is a primary ideal if ''p'' is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity. Examples and properties * The definition can be rephrased in a more symmetric m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all the multiple (mathematics), multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary ideal, primary and semiprime ideal, semiprime. Prime ideals for commutative rings Definition An ideal (ring theory), ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if n\Z is a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull Intersection Theorem
In mathematics, more specifically in ring theory, local rings are certain ring (mathematics), rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place (mathematics), place, or prime. Local algebra is the branch of commutative algebra that studies commutative ring, commutative local rings and their module (mathematics), modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring (mathematics), ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal ideal, maximal left ring ideal, ideal. * ''R'' has a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is affected by touch or other sensation; see Allochiria * Neurologic localization, in neurology, the process of deducing the location of injury based on symptoms and neurological examination * Nuclear localization signal, an amino acid sequence on the surface of a protein which acts like a 'tag' to localize the protein in the cell * Sound localization, a listener's ability to identify the location or origin of a detected sound * Subcellular localization, organization of cellular components into different regions of a cell Engineering and technology * GSM localization, determining the location of an active cell phone or wireless transceiver * Robot localization, figuring out robot's position in an environment * Indoor positioning system, a networ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Power Of A Prime Ideal
In algebra and algebraic geometry, given a commutative Noetherian ring R and an ideal I in it, the ''n''-th symbolic power of I is the ideal : I^ = \bigcap_ \varphi_P^(I^n R_P) where R_P is the localization of R at P, we set \varphi_P : R \to R_P is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of R/I. Though this definition does not require I to be prime, this assumption is often worked with because in the case of a prime ideal, the symbolic power can be equivalently defined as the I - primary component of I^n. Very roughly, it consists of functions with zeros of order ''n'' along the variety defined by I. We have: I^ = I and if I is a maximal ideal, then I^ = I^n. Symbolic powers induce the following chain of ideals: : I^=R\supset I=I^\supset I^\supset I^\supset I^\supset \cdots Uses The study and use of symbolic powers has a long history in commutative algebra. Krull’s famous proof of his princip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Prime Ideal
In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by \operatorname_R(M), and sometimes called the ''assassin'' or ''assassinator'' of (word play between the notation and the fact that an associated prime is an ''annihilator''). In commutative algebra, associated primes are linked to the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal ''J'' is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with \operatorname_R(R/J). Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes. Definitions A nonzero ''R''-module ''N'' is called a prime module if the annihilator \mathrm_R(N)=\mathrm_R(N')\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primal Ideal
In mathematics, an element ''a'' of a commutative ring ''R'' is called (relatively) prime to an ideal ''I'' if whenever ''ab'' is an element of ''I'' then ''b'' is also an element of ''I''. A proper ideal ''I'' of a commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ... ''A'' is said to be primal if the elements that are not prime to it form an ideal. References *. Commutative algebra {{commutative-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of An Ideal
Radical (from Latin: ', root) may refer to: Politics and ideology Politics * Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century * Radical politics, the political intent of fundamental societal change * Radical Party (other), several political parties *Radicals (UK), a British and Irish grouping in the early to mid-19th century * Radicalization *Politicians from the Radical Civic Union Ideologies * Radical chic, a term coined by Tom Wolfe to describe the pretentious adoption of radical causes * Radical feminism, a perspective within feminism that focuses on patriarchy * Radical Islam, or Islamic extremism * Radical Christianity * Radical veganism, a radical interpretation of veganism, usually combined with anarchism * Radical Reformation, an Anabaptist movement concurrent with the Protestant Reformation Science and mathematics Science * Radical (chemistry), an atom, molec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiprime Ideal
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced rings. For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form n\mathbb Z where ''n'' is a square-free integer. So, 30\mathbb Z is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but 12\mathbb Z\, is not (because 12 = 22 × 3, with a repeated prime factor). The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings. Most definitions and assertions in this article appear in and . Definitions For a commutative ring ''R'', a proper ideal ''A'' is a semiprime ideal if ''A'' satisfies either of the following equivalent conditions: * If ''x''''k'' is in ''A'' for some positive integer ''k'' and element ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all the multiple (mathematics), multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary ideal, primary and semiprime ideal, semiprime. Prime ideals for commutative rings Definition An ideal (ring theory), ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if n\Z is a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of An Ideal
Radical (from Latin: ', root) may refer to: Politics and ideology Politics * Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century * Radical politics, the political intent of fundamental societal change * Radical Party (other), several political parties *Radicals (UK), a British and Irish grouping in the early to mid-19th century * Radicalization *Politicians from the Radical Civic Union Ideologies * Radical chic, a term coined by Tom Wolfe to describe the pretentious adoption of radical causes * Radical feminism, a perspective within feminism that focuses on patriarchy * Radical Islam, or Islamic extremism * Radical Christianity * Radical veganism, a radical interpretation of veganism, usually combined with anarchism * Radical Reformation, an Anabaptist movement concurrent with the Protestant Reformation Science and mathematics Science * Radical (chemistry), an atom, molec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
