der Universitat Munchen Set 2
Prof. Dr. B. Pareigis
Problem set for
Advanced Algebra
(5) Let V be a nite dimensional vector space. Let B = (v
i ji =
1;:::;n) be abasis of V and (v
i
ji=1;:::;n) be the dual dual
basis of the dual space V
. Show that P
n
i=1 v
i v
i
2 V V
doesnot depend on the choice of the basis B and that
8v 2V : X
i v
i (v)v
i
=v
holds.
(Hint: Find an isomorphism End (V)
=
V V
and show
that id
V
is mapped to P
n
i=1 v
i v
i
under this isomorphism.)
(6) (a) Let M
R ,
R N, M
0
R , and
R N
0
be R -modules. Showthat the
following isa homomorphismof abelian groups:
:Hom
R (M;M
0
)
Z Hom
R (N;N
0
)3fg 7!f
R
g 2Hom(M
R N;M
0
R N
0
):
(b) Find exampleswhere is not injectiveand where is not
surjective.
(c) Explainwhyfg isadecomposabletensorwhereasf
R g
is not a tensor.
(7) Give a complete proof of Theorem 1.22. In (5) show how
Hom
T
(M:;N:) becomes anS-R -bimodule.
(8) Find an example of M, N 2 K-Mod-K such that M
K N 6
=
N
K M.
(Hint: You may use K := LL,
K M :=
K
K, and N
K := K
K .
Dene a right K-struc ture on M by (m;n)(a;b) := (ma;na)
and aleft K-struc ture on N by (a;b)(r;s):=(br;bs).)
