Summary: Rings and Algebras Problem set #6. Oct. 20, 2011.
1. Which of the following modules are directly indecomposable: ZZ, ZQ, ZR, ZQ[x], ZZp ,
Q[x]Q[x], for V a finite dimensional vector space (V ) (V ), for a finite graph without
oriented cycles KK and KKe1.
2. Which of the following statements are true?
a) The submodule of a directly indecomposable module is directly indecomposable.
b) The homomorphic image of a directly indecomposable module is directly indecomposable.
c) If the modules RM and SN have isomorphic submodule lattices then M is indecomposable
if and only if N is indecomposable.
3. A module M is called uniform if any two nonzero submodules of M have nonzero intersection.
a) Show that the intersection of all nontrivial submodules in a uniform module can be zero.
b) Show that uniform modules are necessarily directly indecomposable but indecomposable
modules are not necessarily uniform.
c) Prove that an injective module is uniform if and only if it is directly indecomposable.
4. a) Show that the composition length of modules is additive in the sense that if M N then
(N) = (M) + (N/M).
b) Prove Fitting's lemma: If M is a module of composition length n < and f is an
endomorphism of M then M = Im fn