structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian...
15 KB (2,160 words) - 18:51, 20 June 2024
by the structure theorem, taking Z as the principal ideal domain. Finitely generated (say left) modules over a division ring are precisely finite dimensional...
19 KB (2,837 words) - 22:51, 25 November 2023
fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal...
12 KB (1,658 words) - 07:13, 19 May 2024
words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal...
21 KB (2,941 words) - 13:38, 7 August 2024
principal ideal domain. The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely generated R-module, then M {\displaystyle...
10 KB (1,446 words) - 11:53, 24 June 2024
Primary decomposition (redirect from Lasker-Noether theorem)
form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it...
26 KB (4,366 words) - 08:47, 7 November 2023
Abelian group (redirect from Fundamental theorem of finite abelian groups)
the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees...
36 KB (5,284 words) - 19:55, 26 August 2024
Torsion (algebra) (redirect from Torsion (modules))
ideal domain and M is a finitely generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed...
12 KB (1,657 words) - 16:54, 17 May 2024
element. Finitely generated ideal: This type of ideal is finitely generated as a module. Primitive ideal: A left primitive ideal is the annihilator of a simple...
37 KB (6,347 words) - 13:52, 10 September 2024
useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated...
24 KB (3,745 words) - 16:27, 10 June 2024
Smith normal form (section Step I: Choosing a pivot)
the structure theorem for finitely generated modules over a principal ideal domain, which includes the fundamental theorem of finitely generated abelian...
16 KB (2,877 words) - 09:45, 29 April 2024
P} is finite is a straightforward application of the structure theorem for finitely generated modules over a principal ideal domain. For modules indexed...
22 KB (2,966 words) - 01:55, 12 December 2023
= K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum...
25 KB (3,937 words) - 20:11, 22 July 2024
Structure theorem may refer to: Structured program theorem, a result in programming language theory Structure theorem for finitely generated modules over...
457 bytes (84 words) - 12:02, 4 January 2022
List of abstract algebra topics (section Module theory)
theorems Composition series Length of a module Structure theorem for finitely generated modules over a principal ideal domain Homological dimension Projective...
12 KB (1,128 words) - 01:18, 14 November 2023
Ring (mathematics) (redirect from Ring with a unit)
finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra. Let V be a finite-dimensional...
99 KB (13,632 words) - 06:07, 27 August 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
free A stably free module structure theorem The structure theorem for finitely generated modules over a principal ideal domain says that a finitely generated...
20 KB (2,611 words) - 13:40, 22 July 2024
Elementary divisors (category Module theory)
a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain....
3 KB (445 words) - 18:44, 15 August 2024
finitely generated modules over a principal ideal domain (abstract algebra) Structure theorem for Gaussian measures (measure theory) Structured program...
73 KB (6,015 words) - 12:17, 2 August 2024
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃...
20 KB (3,124 words) - 14:07, 23 June 2024
Algebra representation (category Module theory)
operator on a finite-dimensional vector space. Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this...
7 KB (1,069 words) - 10:21, 30 June 2021
Polynomial ring (category Free algebraic structures)
The structure theorem for finitely generated modules over a principal ideal domain applies to K[X], when K is a field. This means that every finitely generated...
51 KB (8,173 words) - 08:41, 8 September 2024
theorem for finitely generated modules over a principal ideal domain to V, viewing it as an F[X]-module. The structure theorem provides a decomposition...
16 KB (2,832 words) - 14:45, 31 January 2024
Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity) Structure theorem for finitely generated modules over a principal ideal...
5 KB (654 words) - 14:00, 1 July 2024
direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain § Uniqueness for example. Some authors distinguish...
33 KB (5,663 words) - 12:43, 25 June 2024
Jordan normal form (redirect from Jordan canonical form theorem)
application to the ring K[x] of the structure theorem for finitely generated modules over a principal ideal domain, of which it is a corollary. One can see that...
43 KB (6,836 words) - 12:37, 25 June 2024
Topological data analysis (section Structure theorem)
structure theorem is the standard structure theorem for finitely generated modules over a principal ideal domain. However, this argument fails if the indexing...
89 KB (11,294 words) - 18:07, 8 September 2024
based upon a homotopical theory of modules over highly structured ring spectra. The homotopy category of such modules closely resembles the derived category...
10 KB (1,708 words) - 22:11, 8 April 2024
Invariant factor (category Module theory)
a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain....
1 KB (259 words) - 18:54, 12 August 2023