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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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Abstract Algebra Questions
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