• In category theory, a finitely generated object is the quotient of a free object over a finite set, in the sense that it is the target of a regular epimorphism...
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  • concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional...
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  • finitely generated (or of finite type) over R. The polynomial algebra K [ x 1 , … , x n ] {\displaystyle K[x_{1},\dots ,x_{n}]} is finitely generated...
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  • scheme, a global version of a finitely presented algebra finitely presentable object, in category theory Finitely generated object This disambiguation page...
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  • However, it is true that for finitely presented modules M over a commutative ring R (in particular if M is a finitely generated R-module and R is Noetherian)...
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  • any non-zero finitely generated objects. A Grothendieck category is called locally finitely generated if it has a set of finitely generated generators (i...
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  • -presentable object is called finitely presentable. In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The...
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  • Thumbnail for Monoid
    to generate M if the smallest submonoid of M containing S is M. If there is a finite set that generates M, then M is said to be a finitely generated monoid...
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  • \ldots ,x_{n}\}),} and one says that K ( S ) {\displaystyle K(S)} is finitely generated over K {\displaystyle K} . If S {\displaystyle S} consists of a single...
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  • Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate...
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  • Thumbnail for Finite group
    Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter...
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  • 0 must be finite; Schur's theorem: a torsion linear group is locally finite. In particular, if it is finitely generated then it is finite. Selberg's...
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  • Thumbnail for Convex cone
    polyhedral if it is the conical hull of finitely many vectors (this property is also called finitely-generated). I.e., there is a set of vectors { v 1...
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  • objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness...
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  • locally finite if every finitely generated subgroup is finite, and a group is locally soluble if every finitely generated subgroup is soluble. For finite groups...
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  • a (countable) structure by its finitely generated substructures. Given a class K {\displaystyle \mathbf {K} } of finite relational structures, if K {\displaystyle...
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  • is covered by finitely many affine open sets Spec  A {\displaystyle {\text{Spec }}A} where each A {\displaystyle A} is finitely generated as a B {\displaystyle...
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  • power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets). A topology on a finite set can also...
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  • self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan [de] (1939). Finite differences trace their...
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  • category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. If R is a ring...
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  • Thumbnail for Free group
    finite here, then G is called finitely generated. The kernel Ker(φ) is the set of all relations in the presentation of G; if Ker(φ) can be generated by...
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  • requiring that any finitely generated R-module M is semi-simple. Examples of semi-simple rings include fields and, more generally, finite direct products...
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  • idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by: e...
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  • finitely generated by finitely many invariants of G (because if we are given any – possibly infinite – subset S that generates a finitely generated ideal...
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  • object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects...
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  • Hausdorff. Compact-open topology – Type of topology Countably generated space Finitely generated space – Type of topology in mathematicsPages displaying short...
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  • Thumbnail for Finite element method
    implemented by the construction of a mesh of the object: the numerical domain for the solution that has a finite number of points. FEM formulation of a boundary...
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  • and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an automatic group. Dynkin diagrams have the additional...
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  • Classification of Finitely generated abelian group – Commutative group where every element is the sum of elements from one finite subset Classification...
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  • free. A module M over a ring R is stably free if there exists a free finitely generated module F over R such that M ⊕ F {\displaystyle M\oplus F} is a free...
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