Mathematisches Institut WS 2001/02 der Universität München Set 1 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(1) Let R be a ring and M an abelian group. Show that there is a one-to-one correspondence between maps f : R×M ⟶ M that make M into a left R-module and ring homomorphisms (always preserving the unit element) g : R ⟶ End(M).

(2) Let f : M ⟶ N be an R-module homomorphism. The following are equivalent:

(a) f is a monomorphism,

(b) for all R-modules P and all homomorphisms g, h : P ⟶ M

fg = fh ⟹ g = h,

(c) for all R-modules P the homomorphism of abelian groups

Hom_{R}(P, f) : Hom_{R}(P, M) ∋ g ↦ fg ∈ Hom_{R}(P, N)

is a monomorphism.

(3) Are {(0, …, a, …, 0)|a ∈ K_{n}} and {(a, 0, …, 0)|a ∈ K_{n}} isomorphic as M_{n}(K)-modules?

(4) Show: m : ℤ/(18) ⊗_{ℤ} ℤ/(30) ∋ \overline{x} ⊗ \overline{y} ↦ \overline{xy} ∈ ℤ/(6) is a homomorphism and m is bijective.

Due date: Tuesday, 23.10.2001, 16:15 in Lecture Hall E06