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 RMod, 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 R­Mod, i. e. for each R­module 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 non­zero 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 e 1 , . . . , e n denote the idempotents corresponding to vertices. a) Show that the module Ae i is indecomposable projetive and it is the projective cover of the simple module Ae i /J(A)e i . b) Show that D(e i A) = HomK (e i A, K) is indecomposable injective and it is the injective envelope of Ae i /J(A)e i . 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 : R­Mod # S­Mod is a categorical equivalence then a module RM is faithful if and only if S F (M) is faithful. Derive from this that R is (semi)primitive if and only if S is semiprimitive. 6. Let RPS and S QR be bimodules satisfying the conditions in the the theorem characterizing Morita­equivalence. Prove the natural isomorphisms: Collections: Mathematics