Summary: Rings and Algebras Problem set #6: Solutions Oct. 20, 2011.
1. Which of the following modules are directly indecomposable: Z Z, Z Q, Z R, Z Q[x], Z Z p # ,
Q[x] Q[x], for V a finite dimensional vector space #(V ) # (V ), for # a finite graph without
oriented cycles K#K# and K#K#e 1 .
Solution. Z Z and Z Q are indecomposable because they don't have disjoint nonzero submodules. Z R and Z Q[x]
are decomposable because they decmpose as infinite dimensional Qvectorspaces. Z Zp # 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 zerodivisors). #(V )
# (V ) is indecomposable since it is local: the quotient modulo its
radical is isomorphic to the base field. The path algebra K#K# 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, K#K#e1 is indecomposable since its endomorphism ring is isomorphic to e1K#K#e1 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 S N 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