der Universitat Munchen Set 1
Prof. Dr. B. Pareigis
Problem set for
Advanced Algebra
(1) LetR bea ring and M anabelian group. Show that there isa
one-to-onecorrespondencebetween mapsf :RM !M that
makeM intoaleftR -moduleandringhomomorphisms(always
preserving the unit element)g :R !End (M).
(2) Letf :M !NbeanR -modulehomomorphism. Thefollowing
are equivalent:
(a) f isa monomorphism,
(b) forallR -modulesP andallhomomorphismsg;h:P !M
fg =fh=)g =h;
(c) for all R -modules P the homomorphismof abelian groups
Hom
R
(P;f): Hom
R
(P;M)3g 7!fg 2Hom
R (P;N)
is a monomorphism.
(3) Are f(0;:::;a;:::;0)ja 2 K
n
g and f(a;0;:::;0)ja 2 K
n g iso-
morphicasM
n
(K)-modules?
(4) Show: m : Z=(18)
Z
Z=(30) 3 x y 7! xy 2 Z=(6) is a
homomorphismand m is bijective.
