Abstract Algebra Prelim Jan. 2011
1. Let Rbe a commutative ring with identity, and let Mbe an R-module. Recall the annihilator
of Mis Ann(M) = {r∈R|rm = 0 for all m∈M}. For any ideal Iin R, show Mis an
R/I-module by the rule (r+I)·m=rm if and only if I⊆Ann(M).
2. Let Rbe a commutative ring with identity, and let Iand Jbe ideals in R. Recall that
I+J={r+r0|r∈I, r 0∈J}, and IJ is the ideal generated by all products rr0with r∈I
and r0∈J.
(a) Prove that if I+J=Rthen IJ =I∩J.
(b) Assuming that I+J=R, show that for any aand bin Rthere exists some x∈Rsuch
that x≡amod Iand x≡bmod J. (Recall that x≡amod Iif and only if x−a∈I.)
3. Let ϕ:Z→Aut(Z) by n7→ ϕn, where ϕn(a)=(−1)na. Define the semi-direct product group
G=ZoϕZ.
(a) Write down the group law and the formula for inverses in G.
(b) Find the center of G.
4. In a commutative ring R, an ideal Qis called primary if whenever any aand bin Rsatisfy
ab ∈Qand a6∈ Q, we have bn∈Qfor some integer n≥1. (Equivalently, if ab ≡0 mod Q
and a6≡ 0 mod Q, we have bn≡0 mod Qfor some integer n≥1. That is, in the ring R/Q
any zero divisor is nilpotent.) Show that the nonzero primary ideals in a PID are the ideals of
the form (pn) where pis a prime element and nis a positive integer. You may use that a PID
is a UFD.
5. In R3aline-plane pair is a pair of subspaces (V1, V2) where V1⊂V2, dim V1= 1, and
dim V2= 2. The standard line-plane pair in R3is (Re1,Re1+Re2) where e1= (1,0,0) and
e2= (0,1,0). Let Sbe the set of all line-plane pairs in R3.
(a) The group GL(3,R) of invertible 3 ×3 real matrices acts on Sby
A·(V1, V2) = (A(V1), A(V2)),
where A∈GL(3,R) and (V1, V2)∈ S. Prove that the stabilizer subgroup of the standard
line-plane pair is the group of invertible upper-triangular matrices in GL(3,R) (with
arbitrary non-zero entries on the diagonal).
(b) Prove that the GL(3,R)-action on Sis transitive.
6. Give examples as requested, with brief justification.
(a) A maximal ideal in C[x, y ] which contains the ideal (xy, x2−1).
(b) A ring Rand ideals Iand Jin Rsuch that I J 6=I∩J.
(c) A generator of the group of characters of (Z/7Z)×.
(d) A finite nonzero Z[i]-module.