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...

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...

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...

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

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...

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

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...

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...

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...

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...

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

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...

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

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...

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...

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

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

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

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

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...

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}...

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...

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...

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

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...

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

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...