RepresentationTheory— RevisionSheet0 TU Kaiserslautern
Jun.-Prof. Dr. CarolineLassueur, BernhardB ¨ohmler FB Mathematik
Due date:– WS 2020/21
This sheet is a revision sheet on modules and algebras. These exercises are to be discussed with the assistant during the tutorial on the 29th of October.
Throughout, RandS denote rings. Unless otherwise stated, all rings are assumed to be associative rings with1, and modules are assumed to beleftmodules.
Exercise1 (Change of the base ring).
Prove that ifϕ:S−→Ris a ring homomorphism, then everyR-moduleMcan be endowed with the structure of anS-module with external composition law given by
·:S×M−→M,(s,m)7→s·m:=ϕ(s)·m.
[Hint: use Definition A.1(1) rather than the module axioms in Example A.4(a).]
Exercise2.
Prove that ifKis a field, then anyK-algebra of dimension 2 is commutative.
[Hint: consider aK-basis containing the 1 element.]
Exercise3.
Assume thatRis a commutative ring
(a) LetM,NbeR-modules. Prove that the abelian group HomR(M,N) is a leftR-module for the external composition law defined by
(r f)(m) := f(rm)=r f(m) ∀r∈R,∀f ∈HomR(M,N), ∀m∈M.
(b) LetAbe anR-algebra andMbe anA-module. Prove that EndR(M) and EndA(M) are R-algebras.
Exercise4.
AssumeRis a commutative ring andI is an ideal ofR. LetM be a leftR-module. Prove that there is an isomorphism of leftR-modules
R/I⊗RMM/IM.
