Commutative algebra - Department of Mathematical Sciences - old ...

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Modules of finite length
7.1. Simple Modules
7.1.1. Definition. A nonzero module M is a simple module if 0 and M are the
only submodules.
7.1.2. Proposition. Let f : M → M ′ be a homomorphism
(1) If M is simple, then f is either zero or injective.
(2) If M ′ is simple, then f is either zero or surjective.
(3) If M, M ′ are both simple, then f is either zero or an isomorphism.
Proof. This follows from 2.3.3.
7.1.3. Proposition. A simple module is isomorphic to a factor module R/P for
some uniquely determined maximal ideal P . A simple module is finite, in fact
generated by one element.
Proof. A nonzero f : R → M must be surjective and R/P is simple exactly when
P is maximal. Clearly P = Ann(M).
7.1.4. Corollary. Any nonzero finite module has a simple factor module.
Proof. If M = 0 then MP = 0 for some maximal ideal. By 6.4.1 M ⊗R k(P ) =
M/P M = 0. Choose a nonzero linear projection M/P M → k(P ), giving M →
k(P ) → 0 exact.
7.1.5. Example. (1) If K is a field, a one dimensional vector space is simple.
(2) If R is a principal ideal domain, then R/(p) is a simple module for all irreducible
(p).
(3) Z/(p) are simple for all prime numbers p.
(4) Let K be a field. K[X]/(X − a) is a simple K[X]-module.
7.1.6. Lemma. Let I ⊂ R be an ideal. A simple R/I-module is a simple Rmodule.
7.1.7. Exercise. (1) Show that Z does not contain any simple modules.
(2) Show that Q does not have any simple factor Z-module.
(3) Let L, L ′ ⊂ M be simple submodules. Show that either L ∩ L ′ = 0 of L = L ′ .
(4) Let L = L ′ ⊂ M be simple submodules. Show that L ⊕ L ′ = L + L ′ .
7.2. The Length
7.2.1. Definition. Let R be a ring. A module M is of finite length if it admits a
composition series by submodules
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn−1 ⊂ Mn = M
such that each factor Mi/Mi−1 is a simple module.
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