Math 2.0

Associative rings and algebras Theorem

Jordan–Hölder theorem for modules

Builds on RingModule over a ringSubmoduleLength of a module

Statement

Let RR be a ring and let MM be a left module over RR. A simple module is a nonzero module with no submodules other than 00 and itself. A composition series of MM is a chain of submodules 0=M0⊊M1⊊⋯⊊Mn=M0 = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n = M in which each quotient Mi/Mi−1M_i / M_{i-1} is simple. Let 0=M0⊊⋯⊊Mn=M0 = M_0 \subsetneq \cdots \subsetneq M_n = M and 0=N0⊊⋯⊊Nm=M0 = N_0 \subsetneq \cdots \subsetneq N_m = M be two composition series of MM. Then n=mn = m, and there is a permutation σ\sigma of {1,…,n}\set{1, \dots, n} with Mi/Mi−1≅Nσ(i)/Nσ(i)−1M_i / M_{i-1} \cong N_{\sigma(i)} / N_{\sigma(i) - 1} for each ii.

Remarks

The proof is by induction on the shortest length of a composition series, comparing the two series through the submodules Mn−1M_{n-1}, Nm−1N_{m-1} and Mn−1∩Nm−1M_{n-1} \cap N_{m-1}. So the number nn and the simple factors, counted with multiplicity, depend only on MM; the number nn is the length of MM. For example, ℤ/12ℤ\Z/12\Z has the composition series 0⊂6ℤ/12ℤ⊂3ℤ/12ℤ⊂ℤ/12ℤ0 \subset 6\Z/12\Z \subset 3\Z/12\Z \subset \Z/12\Z and 0⊂6ℤ/12ℤ⊂2ℤ/12ℤ⊂ℤ/12ℤ0 \subset 6\Z/12\Z \subset 2\Z/12\Z \subset \Z/12\Z, with factors ℤ/2ℤ,ℤ/2ℤ,ℤ/3ℤ\Z/2\Z, \Z/2\Z, \Z/3\Z in different orders. A module that has a composition series is exactly a module of finite length, and then every strict chain of submodules can be refined to a composition series. The theorem for groups is the Jordan–Hölder theorem; both are cases of the Jordan–Hölder theorem for groups with operators, and the same proof works.

Going deeper

For a student

The theorem makes the length of a module well defined, and it lets one speak of the multiplicity of a simple module in MM. Lengths of modules such as R/IR/I, for a Noetherian local ring RR, are the input of Hilbert–Samuel functions and multiplicities. In representation theory, the simple factors of a representation are its irreducible constituents, which are well defined by this theorem.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptModule over a ring, definition, no evidence levelModule over a ringdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionRing, definition, no evidence levelRingdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSubmodule, definition, no evidence levelSubmoduledefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSimple group, definition, no evidence levelSimple groupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionComposition series, definition, no evidence levelComposition seriesdefinitionLength of a module, definition, no evidence levelLength of a moduledefinitionJordan–Hölder theorem for modules, statement, no evidence levelJordan–Hölder theorem…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Module over a ring definition
  3. Conjugate elements definition
  4. Ring definition
  5. Subgroup definition
  6. Submodule definition
  7. Conjugate subgroup definition
  8. Left coset definition
  9. Right coset definition
  10. Normal subgroup definition
  11. Simple group definition
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. Subnormal series definition
  15. Composition series definition
  16. Length of a module definition
  17. Jordan–Hölder theorem for modules statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Composition_series
About this page
TagM1ABX18FC4
URIm2://M1ABX18FC4
Kindstatement, theorem
WritingW0 stub
MSC16D10, 13C05
StatusThis copy of the site has no build data. Run m2 status M1ABX18FC4.
AboutLength of a module
UsesRing, Module over a ring, Submodule, Length of a module
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.