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 # , 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 Collections: Mathematics