Abstract Algebra

1. Modules
   1. Definition and Examples
      1. Formal Definition
         1. Modules over a ring
         2. Key properties and axioms
      2. Examples of Modules
         1. \( \mathbb{Z}^n \) as a module over \( \mathbb{Z} \)
         2. Vector spaces as modules over fields
         3. Polynomial modules
         4. Ideals as modules over themselves
   2. Comparison to Vector Spaces
      1. Ring vs Field as Scalar Set
      2. Basis Concepts
         1. Existence of basis
         2. Dependence on ring properties (free vs non-free modules)
      3. Dimension Differences
         1. Vector space dimension is fixed
         2. Module "rank" concept for free modules
   3. Submodules and Direct Sums
      1. Definition of Submodules
         1. Closure under addition
         2. Closure under scalar multiplication
      2. Properties of Submodules
         1. Inclusion relationships
         2. Examples involving specific modules
      3. Direct Sum of Modules
         1. Construction of direct sums
         2. Applications and examples
         3. Internal vs External direct sums
   4. Module Homomorphisms
      1. Definition and Properties
         1. Linear maps in the context of modules
         2. Preservation of module structure
      2. Kernel and Image
         1. Definitions analogously to vector spaces
         2. Relation to submodules
      3. Module Isomorphisms
         1. Definition and examples
         2. Criteria for isomorphism
         3. Relationship to vector spaces
   5. Free Modules
      1. Definition and Characteristics
         1. Modules with a basis
         2. Connection to free abelian groups
      2. Basis and Dimension Theory
         1. Existence of bases
         2. Generating sets and relations
      3. Construction and Examples
         1. Direct product construction
         2. \( R^n \) as a free module over a ring \( R \)
      4. Special Cases and Non-examples
         1. Non-free modules over non-principal ideal domains
   6. Noetherian and Artinian Modules
      1. Noetherian Modules
         1. Definition: ascending chain condition
         2. Examples and properties
         3. Importance in algebraic geometry
      2. Artinian Modules
         1. Definition: descending chain condition
         2. Examples in linear algebra
         3. Connection to rings and field extensions
      3. Relationship Between Noetherian and Artinian
         1. Conditions for being both
         2. Implications of the chain conditions
      4. Applications in Theorems
         1. Hilbert's Basis Theorem
         2. Structure theorems for finitely generated modules