• 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...
    20 KB (2,878 words) - 01:09, 6 May 2025
  • 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...
    23 KB (3,081 words) - 01:20, 19 May 2025
  • theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms. For every finitely generated module M over...
    15 KB (2,160 words) - 10:01, 5 March 2025
  • Finitely generated abelian group Finitely generated module Finitely generated ideal Finitely generated algebra Finitely generated space This disambiguation page...
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  • 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...
    8 KB (1,115 words) - 23:54, 13 May 2025
  • 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...
    9 KB (1,182 words) - 08:57, 7 November 2023
  • have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated modules...
    13 KB (2,157 words) - 23:31, 13 May 2025
  • 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...
    3 KB (490 words) - 00:05, 4 June 2025
  • a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over...
    26 KB (4,366 words) - 02:50, 26 March 2025
  • 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...
    30 KB (4,590 words) - 03:05, 9 August 2024
  • 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...
    66 KB (9,772 words) - 22:19, 27 May 2025
  • 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...
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  • 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...
    22 KB (3,091 words) - 12:09, 26 March 2025
  • with finitely generated modules: a submodule of a finitely generated module need not be finitely generated. The integers, considered as a module over...
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  • 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...
    3 KB (563 words) - 02:35, 13 May 2024
  • infinite. Finitely-generated modules over principal ideal domains (PIDs) are classified by the structure theorem for finitely generated modules over a principal...
    5 KB (705 words) - 22:27, 28 October 2023
  • 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...
    13 KB (2,160 words) - 20:22, 18 October 2024
  • 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...
    24 KB (3,715 words) - 01:53, 1 June 2025
  • ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative...
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  • Thumbnail for Finitely generated group
    cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite. Finitely generated module Presentation of...
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  • a finite (hence finitely generated) abelian group. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian...
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  • Press. p. 21. ISBN 9780201407518. Finitely generated module Finitely generated field extension Artin–Tate lemma Finite algebra Morphism of finite type...
    7 KB (1,207 words) - 15:47, 19 December 2024
  • 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...
    20 KB (2,611 words) - 18:28, 4 March 2025
  • 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}...
    13 KB (2,281 words) - 15:01, 11 January 2025
  • 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...
    12 KB (1,660 words) - 18:12, 1 December 2024
  • 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...
    3 KB (404 words) - 00:12, 27 April 2024
  • Thumbnail for Abelian group
    classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal...
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  • Thumbnail for Zero object (algebra)
    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...
    8 KB (939 words) - 19:17, 5 January 2025
  • a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a Dedekind domain, a finitely generated torsion-free...
    5 KB (658 words) - 10:48, 9 February 2024
  • 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...
    38 KB (6,198 words) - 10:42, 15 May 2025