RepresentationTheory— ExerciseSheet1 TU Kaiserslautern
Jun.-Prof. Dr. CarolineLassueur FB Mathematik
BernhardB ¨ohmler
Due date:Wednesday, 11th of November 2020, 10 a.m. WS 2020/21
Throughout, R denotes a ring, and, unless otherwise stated, all rings are assumed to be associative rings with1, and modules are assumed to beleftmodules.
A. Exercises for the tutorial.
Exercise1.
Prove that if (R,+,·) is a ring, then R◦ := R itself may be seen as an R-module via left multiplication inR, i.e. where the external composition law is given by
R×R◦−→R◦,(r,m)7→r·m. We callR◦theregularR-module.
Prove that:
(a) theR-submodules ofR◦are prescisely the left ideals ofR;
(b) I/Ris a maximal left ideal ofR⇔R◦/Iis a simpleR-module, andI/Ris a minimal left ideal ofR⇔Iis simple when regarded as anR-submodule ofR◦.
Exercise2.
(a) Give a concrete example of anR-module which is indecomposable but not simple.
(b) Prove that any submodule and any quotient of a semisimple module is again semisim- ple.
Exercise3.
Prove Part (iii) of Fitting’s Lemma.
B. Exercises to hand in.
Exercise4.
(a) Letpbe a prime number andR:={a
b ∈Q
p-b}. Prove thatR\R×={a
b ∈R
p|a}and deduce thatRis local.
(b) LetKbe a field and letR:=n A=
a1 a2 ... an
0 a1 ... an−1
... ... ...
0 0 ... a1
∈Mn(K)o . Prove thatR\R×={A∈R|a1 =0}and deduce thatRis local.
Exercise5.
(a) Prove that any simpleR-module may be seen as a simpleR/J(R)-module.
(b) Conversely, prove that any simpleR/J(R)-module may be seen as a simpleR-module.
[Hint: use a change of the base ring via the canonical morphismR−→R/J(R).]
(c) Deduce thatRandR/J(R) have the same simple modules.
Exercise6.
(a) LetKbe a field and letAbe theK-algebrana1 a
0 a1
|a1,a∈Ko
. Consider theA-module V:=K2, whereAacts by left matrix multiplication. Prove that:
(1) {(x0)|x∈K}is a simpleA-submodule ofV; but (2) Vis not semisimple.
(b) Prove thatJ(Z)=0 and find an example of aZ-module which is not semisimple.
