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    UCSD Ring and Module Theory Qual Review

    SRC-UCSD-ALG-REVIEW-RINGS-MODULES

    Area
    Algebra
    Status
    draft
    Source
    Contents

    8 problems.

    Spring 2007, problems 2–3

    1. Intersection of ideals and generators in a PID
    2. All R[x]-module structures on a 3-dimensional real vector space

    Fall 2009, problem 8

    1. Intersection of noetherian ring quotients is noetherian

    Spring 2012, problems 3 and 5

    1. Surjective endomorphisms of Noetherian rings are injective
    2. Polynomials that force diagonalizability over an algebraically closed field

    Fall 2009, problems 2–4 (labels printed on the review sheet)

    1. Examples of nilpotents in quotient rings versus base rings
    2. Ring homomorphism from C[x] onto C and non-surjective examples
    3. Tensoring finitely generated modules with \(\mathbb Q\) and extending scalars \(\mathbb R[x]\to\mathbb C[x]\)

    Browse these problems in source order