a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module...

a finitely generated projective A-module containing A as a subring, then A is a direct factor of B. Let P be a finitely generated projective module over...

theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms. For every finitely generated module M over...

sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion module, which happens...

Finitely generated abelian group Finitely generated module Finitely generated ideal Finitely generated algebra Finitely generated space This disambiguation page...

and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both...

have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated modules...

a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over...

infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group). A finitely generated module over a commutative local ring is...

f:F\to M,} where F {\displaystyle F} is a finitely generated free R-module, and for every finitely generated R-submodule K {\displaystyle K} of ker ⁡ f...

over itself. If there is a finite generating set, then a module is said to be finitely generated. This applies to ideals, which are the submodules of the...

with finitely generated modules: a submodule of a finitely generated module need not be finitely generated. The integers, considered as a module over...

Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator...

much weaker than saying it is finitely generated as a module. 3.  An extension of fields is called finitely generated if elements of the larger field...

ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative...

particular, P {\displaystyle P} is a finitely generated free module. Now let M {\displaystyle M} be a finitely generated module over an arbitrary Dedekind domain...

denoted by R-Mod (see category of modules). Finitely generated An R-module M is finitely generated if there exist finitely many elements x1, ..., xn in M...

of the standard basis generates M. In particular, if J is finite, then M is a finitely generated module. If I and J are finite sets, then the presentation...

a finite (hence finitely generated) abelian group. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian...

are finitely generated free modules. 2.  A finitely presented module is a module that admits a finite free presentation. finitely generated A module M {\displaystyle...

infinite. Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principal...

Press. p. 21. ISBN 9780201407518. Finitely generated module Finitely generated field extension Artin–Tate lemma Finite algebra Morphism of finite type...

cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite. Finitely generated module Presentation of...

more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization...

principal ideal domain and M is a finitely generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives...

classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal...

a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a Dedekind domain, a finitely generated torsion-free...

free resolution. Hilbert's syzygy theorem states that, if M is a finitely generated module over a polynomial ring k [ x 1 , … , x n ] {\displaystyle k[x_{1}...

κ ∈ R. The most general of them, the zero module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication...

nonzero finitely generated module admits a maximal submodule, in particular, one has: If J M = M {\displaystyle JM=M} and M is finitely generated, then...