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...

concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional...

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...

scheme, a global version of a finitely presented algebra finitely presentable object, in category theory Finitely generated object This disambiguation page...

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)...

any non-zero finitely generated objects. A Grothendieck category is called locally finitely generated if it has a set of finitely generated generators (i...

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...

-presentable object is called finitely presentable. In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The...

\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...

Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate...

Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter...

objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness...

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...

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...

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...

a (countable) structure by its finitely generated substructures. Given a class K {\displaystyle \mathbf {K} } of finite relational structures, if K {\displaystyle...

object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects...

finitely generated by finitely many invariants of G (because if we are given any – possibly infinite – subset S that generates a finitely generated ideal...

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...

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...

self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan [de] (1939). Finite differences trace their...

requiring that any finitely generated R-module M is semi-simple. Examples of semi-simple rings include fields and, more generally, finite direct products...

Hausdorff. Compact-open topology – Type of topology Countably generated space Finitely generated space – Type of topology in mathematicsPages displaying short...

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...

idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by: e...

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...

see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three...

canonical ring is finitely generated, the canonical model is Proj of the canonical ring. If the canonical ring is not finitely generated, then Proj R is...

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...

Classification of Finitely generated abelian group – Commutative group where every element is the sum of elements from one finite subset Classification...