Abstract Algebra I Final Exam December 13, 2011
MATH 601 R. Hammack
Directions: Solve five of the following ten questions.
A. Groups
1. Recall that if H≤G, then [G, H ] is the subgroup of Ggenerated by all commutators [g, h] =
g−1h−1gh with g∈Gand h∈H.
Prove that His normal in Gif and only if [G, H]≤H.
2. Suppose Aand Bare normal subgroups of Gsuch that G/A and G/B are both abelian.
Prove that G/(A∩B) is abelian.
3. The matrix 0−1
1 4 has order 5 in GL2(F19). Use it to construct a non-abelian group of order
1805.
B. Rings
4. Let Rbe a ring with 1. Arguing strictly from the definition of a ring, show that (−1)2= 1.
5. Let Rbe a ring with 1. Prove that the center of the ring M2(R) is Z= r0
0r|r∈Z(R).
6. The characteristic of a ring Ris the smallest positive integer psuch that 1 + 1 + 1 + · · · + 1 = 0
(ptimes) in R. If no such integer exists, then we say Rhas characteristic 0.
Prove that if an integral domain has characteristic p, then pis either prime or zero.
7. Recall that an element ain a ring Ris nilpotent if an= 0 for some integer n.
Let Rbe a commutative ring with 1 6= 0.
Prove that if ais nilpotent, then 1 −ab is a unit for every b∈R.
8. Suppose Rand Sare rings with identities.
Prove that every ideal of R×Sis of form I×J, where Iis an ideal in Rand Jis an ideal in S.
9. Prove that the quotient ring Z[i]/I is finite for every non-zero ideal Iin the Gaussian integers Z[i].
10. Prove that the quotient of a PID by a prime ideal is again a PID.