Finitely generated algebraIn mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K. Equivalently, there exist elements s.t. the evaluation homomorphism at is surjective; thus, by applying the first isomorphism theorem, . Conversely, for any ideal is a -algebra of finite type, indeed any element of is a polynomial in the cosets with coefficients in .
Noetherian ringIn mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated.
Radical of a moduleIn mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(M) of M. Let R be a ring and M a left R-module. A submodule N of M is called maximal or cosimple if the quotient M/N is a simple module. The radical of the module M is the intersection of all maximal submodules of M, Equivalently, These definitions have direct dual analogues for soc(M).
Spectrum of a ringIn commutative algebra, the prime spectrum (or simply the spectrum) of a ring R is the set of all prime ideals of R, and is usually denoted by ; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings . For any ideal I of R, define to be the set of prime ideals containing I. We can put a topology on by defining the to be This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For f ∈ R, define Df to be the set of prime ideals of R not containing f.
Isomorphism theoremsIn mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
Ring (mathematics)In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Quotient ringIn ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations.
Ring homomorphismIn 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 f : R → S such that f is: addition preserving: for all a and b in R, multiplication preserving: for all a and b in R, and unit (multiplicative identity) preserving: 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.
RAF GrangemouthRoyal Air Force Grangemouth or more simply RAF Grangemouth is a former Royal Air Force station located north east of Falkirk, Stirlingshire, Scotland. It was opened as the Central Scotland Airport in May 1939 and operated as a Civilian Air Navigation School (CANS) until early September 1939. After the war it was used as Gliding School until 1946. It was then used by RAF Maintenance Command until its closure in June 1955.
Martha Reeves (anchorite)Martha Reeves (born 1941) is a vowed Anglican solitary (or anchorite), with Rowan Williams, the former Archbishop of Canterbury, as bishop-protector. A graduate of the Madeira School (Class of 1959), she is also a Stanford-educated professor of theology who has written numerous articles and books under the name "Maggie Ross" as well as translated a number of Carthusian Novice Conferences. Reeves, at one time Desmond Tutu's spiritual director, was Bell Distinguished Professor in Anglican and Ecumenical Studies appointed to the Department of Philosophy and Religion, Kendall College of Arts and Sciences, The University of Tulsa.
NK KarlovacNK Karlovac 1919 is a Croatian football club based in the town of Karlovac. Karlovac plays their home matches at Stadion Branko Čavlović-Čavlek. Traditionally lower-level minnows, the club's most successful period in the Yugoslav football league system was in the 1970s when they competed in the Yugoslav Second League. After the breakup of Yugoslavia and the formation of the Croatian football league system in 1992 Karlovac spent most of the time playing in Druga HNL and Treća HNL, second and third levels.
Sylvain LéviSylvain Lévi (March 28, 1863 – October 30, 1935) was an influential French orientalist and indologist who taught Sanskrit and Indian religion at the École pratique des hautes études. Lévi's book Théâtre Indien is an important work on the subject of Indian performance art, and Lévi also conducted some of the earliest analysis of Tokharian fragments discovered in Western China. Lévi exerted a significant influence on the life and thought of Marcel Mauss, the nephew of Émile Durkheim.
