Transcribed TextTranscribed 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.)

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Abstract Algebra Questions
    $60.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Abstract Algebra Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats