Math 620 Homework 1 Due Monday, September 15, 2003
1. Find the prime ideals of the ring R=๎ZpZ
Z Z ๎.
2. Let M=RMbe a left R-module, and let U=RUSbe an R-S-bimodule.
(a) Show that HomR(U, M) has a natural structure as a left S-module, defined by (sf)(u) = f(us),
for all sโSand all fโHomR(U, M ).
(b) Using this structure on HomR(R, M ), prove that HomR(R, M )โผ
=M(as left R-modules).
3. Let Kbe a field, and let R=๎K0
K K ๎.
(a) Show that the following are simple left R-modules.
S1=๎K0
K K ๎๎๎ 0 0
K K ๎,S2=๎0 0
0K๎,S3=๎0 0
K0๎
(b) Show that S16โผ
=S2, and S2โผ
=S3.
4. Let Sbe a commutative ring, and let Mbe an S-module. We define the trivial extension of Mby S
to be the set of all formal matrices
R=๎๎ s0
m s ๎๎
๎
๎
๎
sโSand mโM๎.
The notation R=Sร
|Mis used, and this ring is sometimes called the idealization of M.
(a) Show that Ris a commutative ring.
(b) If Nis an S-submodule of M, and Iis an ideal of Ssuch that IM โN, we define
Iร
|N=๎๎ a0
x a ๎๎
๎
๎
๎
aโIand xโN๎.
Show that Iร
|Nis an ideal of R.
(c) Show that the correspondence Nโ(0) ร
|Ndetermines a one-to-one correspondence between the
set of S-submodules Nof Mand the set of ideals of Rthat are contained in (0) ร
|M.