PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Algebra → Module Theory ↓


Injective Modules


In module theory, injective modules form an important place for understanding how modules interact within algebras. The injective module concept is full of intrinsic properties that reveal intriguing relations and simplifications within ring theory and module theory. We will explore these structures in comprehensive, plain-language descriptions and examples. We will clarify the notions through examples, properties, and visual demonstrations.

Understanding the module

To dive into injective modules, a basic understanding of modules is required. Modules are a generalization of vector spaces where instead of a field, we have a ring. Suppose R is a ring and M is an abelian group. We say that M is a module over R if there is a multiplication:

R × M -> M

This multiplication must satisfy certain axioms like associativity, distributivity, etc.

Defining injective modules

A module I is called injective if it possesses a specific extension property related to homeomorphisms: every homeomorphism defined on a submodule of any module M can be extended to a homeomorphism on all of M, when mapping it into I

F: N -> I extend to G: M -> I, for N ⊆ M

This property provides deep insight into the structure of modules and often simplifies complex ideas within module theory.

Injective modules - properties and examples

Basic properties of injective modules

  • Every injective module I over a ring R is a direct sum of any modules containing I as submodules.
  • Injective modules have a Baire norm, which essentially states a compositional property of the module extension.
  • For any module M, we can find an injective module I such that M is a submodule of I; such a module is called the injective hull of M

Visual example: homomorphism extension

Let's look at how injective modules work with respect to the extension of homeomorphisms:

N F : N -> I I G : M -> I (Expansion) M I

Example: Integers and Rational Numbers

Consider the ring of integers Z and the additive group of rational numbers Q Here, Q is an injective module over the ring of integers Z Any homomorphism from a subgroup of integers (such as the multiples of an integer) to Q can be extended to a homomorphism from the entire group of integers Z to Q

For example, suppose H is the subgroup of Z consisting of all multiples of 2. Define an isomorphism h: H -> Q by h(2n) = n. This isomorphism can be extended to all of Z:

f(n) = n/2 for all n in Z

This exhibits the injective property, since Q allows this extension.

Construction of injective modules

Constructing injective modules is important for deeper exploration. One way to concretely represent injective modules is through the notion of an injective hull or injective envelope.

The injective hull of a module M, denoted E(M), is the smallest injective module that contains M as a submodule. Technically, this involves a specific choice of embedding and the minimal nature of the inclusion.

Beyond basic rings: examples in endomorphism rings

Consider a ring R as an endomorphism of a vector space V Here, injective modules appear in sophisticated forms via modules such as M = Hom(V, V), where V is naturally extended under the projecting homomorphism definitions.

The Beyer criterion

The Baire criterion provides another way to identify injective modules. It states that a module M R is injective if and only if for every left ideal I of R, every homomorphism from I to M can be extended to a homomorphism from R to M

map: I -> M extends to R -> M

Additional topics and concluding remarks

Applications and theory building

Injective modules find numerous applications in many advanced areas such as homological algebra, category theory, and representation theory. The concepts and results are foundational in understanding the complexity and structure of mathematical objects defined over rings.

Comparison with other modules

When compared with projective modules, the nature of embeddings and extensions provides a dual notion. While projective modules help cover or map modules, injective modules complement them by facilitating extensions.

Conclusion

Injective modules play a key role in understanding how modules behave over rings. Their utility in theory and practice opens the door to concrete results within abstract algebra. Exploring injective properties serves not only to illuminate singular modules but also to enhance the overall understanding of module theory.


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Injective Modules

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