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. 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 : I 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 Aei/J(A)ei. b) Show that D(eiA) = HomK(eiA, K) is indecomposable injective and it is the injective envelope of Aei/J(A)ei. 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. Collections: Mathematics