Summary: Rings and Algebras Problem set #11. Dec. 1, 2011.
1. Suppose e R is a basic idempotent in a semiperfect ring R and suppose that M is a generator of R-Mod, i. e.
for each R-module N there is an epimorphism f :
M N. Then Re is isomorphic to a direct summand of M.
2. Let R be the ring of all N × N matrices over R which can be written as the sum of a scalar matrix and a strictly
lower triangular matrix with only finitely many non-zero entries. Show that R is left perfect but not right perfect.
3. Let A = K/I be a finite dimensional path algebra defined by relations. Let e1, . . . , en denote the idempotents
corresponding to vertices.
a) Show that the module Aei is indecomposable projetive and it is the projective cover of the simple module
b) Show that D(eiA) = HomK(eiA, K) is indecomposable injective and it is the injective envelope of
4. a) Show that the following are equivalent for a module RM:
(i) M is faithful;
(ii) M cogenerates RR;
(iii) M cogenerates every finitely generated projective module.
5. Show that if F : R-Mod S-Mod is a categorical equivalence then a module RM is faithful if and only if SF(M)
is faithful. Derive from this that R is (semi)primitive if and only if S is semiprimitive.
6. Let RPS and SQR be bimodules satisfying the conditions in the the theorem characterizing Morita-equivalence.