## Transcribed Text

1. (21 total points) Let R be a commutative ring with 1. Let N(R) = {a € R|a" = 0 for some n € N}.
(a) (8 points) Prove that N(R) <.R. (Hint: it may be convenient to prove N is closed under addition,
additive inverses and multiplication as opposed to subtraction and multiplication)
(b) (7 points) Let S = R/N(R). Prove that N(S) = {0 + N(R)}.
(c) (6 points) Show that N(R) C P for every prime ideal P <1 R.
2. (15 total points) In this problem, G is an abelian group.
(a) (5 points) Characterize the natural numbers n for which the following statement must be true.
If IG| = n, then G is cyclic.
(b)
(5 points) Characterize the natural numbers n for which the following statement must be true.
If IG = n, then there are exactly 4 choices for the isomorphism type of G.
(c) (5 points) Up to isomorphism, how many choices are there for G, if G must satisfy both of the
following statements?
G = 16 and
for all x € G, |x| divides 4.
3. (20 total points) Prove the following. (Parts (a), (b) and (c) can be treated as separate)
(a) (6 points) Let R be a ring, and let X be a set with two elements. Prove that R x R = S where
S = {f : X
R}. (You do not have to show that S is a ring under pointwise addition and
multiplication)
(b) (5 points) (x) is a prime ideal in Z[x] which is not maximal.
(c) (4 points) Construct an example of a ring homomorphism f : R
S such that all the following
hold.
R and S are both commutative rings with 1.
f is one-to-one.
f(1) is not a unit.
(d) (5 points) Show that {(r,0) r € R} is a maximal ideal in R x Q.
4. (8 total points) Show by example that the following statement is false.
If N, H, G are groups, then if N < H and H 5.
(20 total points) Prove the following (parts (a) and (b) can be treated as separate)
(a) that (10 points) (H xH)D=H. Let H be any abelian group (possibly infinite), and let D = {(h,h) E H x H}. Prove
(b) for (10 points) Let N, for G = (C*,-) Prove that G has no maximal subgroups. (You can use for free that
any n E every z € C, there is a W € C with wn = z.)

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