Mathematisches Institut WS 2001/02 der Universität München Set 6 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(21) Let R := K × K for K a field.

(a) Show that P := {(a, 0)|a ∈ K} is a finitely generated projective R-module.

(b) Decide if the R-modules P and Q := {(0, a)|a ∈ K} are isomorphic?

(c) Find a dual basis for P.

(22) (a) Let R be a ring and P_{R} be an R-module. Show that P is a finitely generated projective module if and only if P is a direct summand of the R-module R^{n}.

(b) Let P_{R} be a finitely generated projective right R-module. Show that P^{∗} = Hom_{R}(P., R.) is a finitely generated projective left R-module.

(23) Let R be a ring. Show that for each projective R-module P there is a free R-module F such that P ⊕ F ≅ F.

(24) Let C be a category with finite products. For each object A in C show that there exists a morphism Δ_{A} : A ⟶ A × A satisfying p_{1}Δ_{A} = 1_{A} = p_{2}Δ_{A}. Show that this defines a natural transformation. What are the functors?

Due date: Tuesday, 27.11.2001, 16:15 in Lecture Hall 138