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...

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...

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...

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...

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

finitely-generated free module and an ideal, and the class of the ideal is uniquely determined by the module. Over a principal ideal domain, finitely-generated...

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

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

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...

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...

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...

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...

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

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

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...

be coherent, since finitely generated projective modules are finitely related. Though ideals of Dedekind domains can all be generated by two elements, for...

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

if M is finitely generated, then the syzygy module is also finitely generated. A syzygy module of this syzygy module is a second syzygy module of M. Continuing...

M, rad(M/rad(M)) is zero. M is a finitely generated module if and only if the cosocle M/rad(M) is finitely generated and rad(M) is a superfluous submodule...

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

any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian...

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...

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

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...