Projective and injective modules are key players in homological algebra. They're like the superheroes of module theory, swooping in to solve problems and make life easier for algebraists.

This section dives into characterizing these modules and provides examples. We'll see how projective and injective modules relate to other concepts like flat modules, vector spaces, and abelian groups.

Projective and Injective Modules

Properties and Constructions of Projective and Injective Modules

• Projective cover
  • Smallest projective module that maps onto a given module
  • Unique up to isomorphism
  • Can be used to measure the complexity of a module
• Injective hull
  • Smallest injective module containing a given module as a submodule
  • Unique up to isomorphism
  • Dual notion to the projective cover
  • Can be constructed using the injective envelope

Homological Dimensions

• Projective dimension
  • Shortest length of a projective resolution of a module
  • Measures how far a module is from being projective
  • Projective modules have projective dimension 0
  • Used to define the global dimension of a ring
• Injective dimension
  • Shortest length of an injective resolution of a module
  • Measures how far a module is from being injective
  • Injective modules have injective dimension 0
  • Used to define the global dimension of a ring

Special Module Types

Flat and Cotorsion Modules

• Flat module
  • Module $M$ such that the functor $- \otimes_R M$ is exact
  • Generalizes the notion of a flat ring homomorphism
  • Projective modules are flat, but the converse is not always true
  • Example: Over a principal ideal domain, a module is flat if and only if it is torsion-free
• Cotorsion module
  • Module $C$ such that $\text{Ext}^1_R(F, C) = 0$ for all flat modules $F$
  • Dual notion to flat modules
  • Injective modules are cotorsion, but the converse is not always true

Vector Spaces

• Vector space
  • Module over a field
  • Has a basis, allowing for coordinate representation of elements
  • Dimension of a vector space is the cardinality of its basis
  • Example: $\mathbb{R}^n$ is a vector space over $\mathbb{R}$ with the standard basis ${e_1, \ldots, e_n}$

Ring Properties

Finiteness Conditions on Rings

• Noetherian ring
  • Ring in which every ideal is finitely generated
  • Equivalent to the ascending chain condition on ideals
  • Examples: Fields, principal ideal domains, and polynomial rings over Noetherian rings
• Artinian ring
  • Ring in which every descending chain of ideals stabilizes
  • Implies the ring is Noetherian
  • Examples: Fields and finite-dimensional algebras over fields

Abelian Groups

Examples and Properties

• Abelian group examples
  • $\mathbb{Z}$ under addition
  • $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ under addition
  • $\mathbb{Z}/n\mathbb{Z}$ (cyclic groups) under addition
  • The multiplicative group of nonzero elements of a field
• Abelian groups are $\mathbb{Z}$-modules
  • The ring action is given by integer multiplication
  • Allows for the application of module-theoretic concepts to abelian groups
• Finitely generated abelian groups are direct sums of cyclic groups
  • Fundamental theorem of finitely generated abelian groups
  • Classification up to isomorphism