“Exercises on Rings and Modules” contains a collection of exercises designed to help students understand and practice concepts related to rings and modules in abstract algebra. Here are the key sections and types of exercises included:

  1. Basic Definitions and Examples:
    • Exercises on defining rings, subrings, ring homomorphisms, and ideals.
    • Problems that require identifying examples and counterexamples of these structures.
  2. Ring Properties:
    • Questions about properties of rings such as commutativity, units, and zero-divisors.
    • Exercises on proving whether certain sets with given operations form a ring.
  3. Ideals and Quotient Rings:
    • Problems involving the definition and properties of ideals, principal ideals, and maximal ideals.
    • Exercises on constructing quotient rings and working with their properties.
  4. Ring Homomorphisms:
    • Exercises on defining and finding examples of ring homomorphisms and isomorphisms.
    • Problems on kernel and image of homomorphisms and their relationship with ideals.
  5. Modules over a Ring:
    • Basic exercises on defining modules, submodules, and module homomorphisms.
    • Problems that involve checking whether given sets with operations form modules over a specified ring.
  6. Exact Sequences and Module Homomorphisms:
    • Exercises on exact sequences, including short exact sequences and their properties.
    • Problems related to the image, kernel, and cokernel of module homomorphisms.
  7. Free Modules and Bases:
    • Problems that require understanding and constructing free modules.
    • Exercises on finding bases of free modules and understanding their properties.
  8. Tensor Products and Exact Sequences:
    • Advanced exercises on the construction and properties of tensor products of modules.
    • Problems that involve exact sequences in the context of tensor products and their applications.
  9. Projective, Injective, and Flat Modules:
    • Exercises on defining and working with projective, injective, and flat modules.
    • Problems related to their characterizations and examples.