Summary: Rings and Algebras Problem set #6: Solutions 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.
Solution. ZZ and ZQ are indecomposable because they don't have disjoint nonzero submodules. ZR and ZQ[x]
are decomposable because they decmpose as infinite dimensional Q-vectorspaces. ZZp is indecomposable
because it has a unique smallest nonzero submodule. Q[x]Q[x] is indecomposable because there are no disjoint
ideals in it (Q[x] has no zero-divisors). (V ) (V ) is indecomposable since it is local: the quotient modulo its
radical is isomorphic to the base field. The path algebra KK is indecomposable if and only if the graph
has one vertex, since the idempotents corresponding to the vertices give a decomposition of the path algebra.
Finally, KKe1 is indecomposable since its endomorphism ring is isomorphic to e1KKe1 and this is a local
ring: actually, paths of nonzero length which start and end at 1 do not exist, thus the endomorphism ring is
the span of e1, hence it is one dimensional.
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.
Solution. a) The statement is false. Take the algebra A =
K 0 0