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Going Up And Going Down
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions. The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going down refers to the case when a chain can be extended by "downward inclusion". The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Seidenberg. These are known as the going-up and going-down theorems. Going up and going down Let ''A'' ⊆ ''B'' be an extension of commutative rings. The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in ''B'', each member of which lies over members of a longer chain of prime ideals in ''A'', to be able to be extended to the length of the chain of prime ideals in ''A''. Lying over and incomparability First, we fix some terminology. If \mathfrak and \mathfrak are prime ideals of ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a 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 properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews. http://www-history.mcs.st-andrews.ac.uk/Biographies/Kaplansky.html. Biography Kaplansky or "Kap" as his friends and colleagues called him was born in Toronto, Ontario, Canada, to Polish-Jewish immigrants; his father worked as a tailor, and his mother ran a grocery and, eventually, a chain of bakeries. He went to Harbord Collegiate Institute receiving the Prince of Wales Scholarship as a teenager. He attended the University of Toronto as an undergraduate and finished first in his class for three consecutive years. In his senior year, he competed in the first William Lowell Putnam Mathematical Competition, becoming one of the first five recipients of the Putnam Fellowship, which paid for graduate studies at Harvard University. Administe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Life Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at the University of Oxford and the University of Cambridge and in the United States at the Institute for Advanced Study. He was the President of the Royal Society (1990–1995), founding director of the Isaac Newton Institute (1990–1996), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and the President of the Royal Society of Edinburgh (2005–2008). From 1997 until his death, he was an honorary professor in the University of Edinburgh. Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch and Isadore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective Function
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Extension
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion Map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota(x)=x. A "hooked arrow" () is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (However, some authors use this hooked arrow for any embedding.) This and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f between objects X and Y, if there is an inclusion map into the domain \iota : A \to X, then one can form the restriction f \, \iota of f. In many instances, one can also construct a canonical inclusion into the codomain R \to Y known as the range of f. Applications of inclusion maps Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserving: ::f(a+b)=f(a)+f(b) for all ''a'' and ''b'' in ''R'', :multiplication preserving: ::f(ab)=f(a)f(b) for all ''a'' and ''b'' in ''R'', :and unit (multiplicative identity) preserving: ::f(1_R)=1_S. Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. If in addition ''f'' is a bijection, then its inverse ''f''−1 is also a ring homomorphism. In this case, ''f'' is called a ring isomorphism, and the rings ''R'' and ''S'' are called ''isomorphic''. From the standpoint of ring theory, isomorphic rings cannot be distinguished. If ''R'' and ''S'' are rngs, then the cor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Extension
In commutative algebra, a ring extension is a ring homomorphism R\to S of commutative rings, which makes an -algebra. In this article, a ring extension of a ring ''R'' by an abelian group ''I'' is a pair of a ring ''E'' and a surjective ring homomorphism \phi:E\to R such that ''I'' is isomorphic (as an abelian group) to the kernel of \phi. In other words, :0 \to I \to E \overset R \to 0 is a short exact sequence of abelian groups. (This makes ''I'' a two-sided ideal of ''E''.) Given a commutative ring ''A'', an ''A''-extension is defined in the same way by replacing "ring" with " algebra over ''A''" and "abelian groups" with "''A''-modules". An extension is said to be ''trivial'' if \phi splits; i.e., \phi admits a section that is an algebra homomorphism. This implies that ''E'' is isomorphic to the direct product of ''R'' and ''I''. A morphism between extensions of ''R'' by ''I'', over say ''A'', is an algebra homomorphism ''E'' → ''E'' that induces the identities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Seidenberg
Abraham Seidenberg (June 2, 1916 – May 3, 1988) was an American mathematician. Early life Seidenberg was born on June 2, 1916 to Harry and Fannie Seidenberg in Washington D.C. He graduated with a B.A. from the University of Maryland in 1937. He completed his Ph.D. in mathematics from Johns Hopkins University in 1943. His Ph.D. thesis, written under the direction of Oscar Zariski, was on ''Valuation Ideals in Rings of Polynomials in Two Variables''. Academic career Seidenberg became an instructor in mathematics at the University of California, Berkeley in 1945. He reached the rank of full professor in 1958. He retired from Berkeley in 1987. Contributions Seidenberg was known for his research in commutative algebra, algebraic geometry, differential algebra, and the history of mathematics. He published ''Prime ideals and integral dependence'' written jointly with Irvin Cohen, which greatly simplified the existing proofs of the going-up and going-down theorems of ideal the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
