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 → Ring Theory ↓


Artinian Rings


In ring theory, a branch of abstract algebra, Artinian rings play a key role in understanding algebraic structures by providing the foundation for various algebraic concepts and theorems. Named after the Austrian mathematician Emil Artin, these rings are specifically defined by their descending chain condition on ideals. Before delving deeper into Artinian rings, it is important to briefly discuss some basic concepts of ring theory, such as the definition of rings, ideals, and other relevant terminology.

Basic definitions and concepts

A ring is a mathematical structure consisting of a set equipped with two binary operations: addition and multiplication. The set is closed under these operations and obeys certain rules. To be more precise, a ring ( R ) must satisfy the following:

  • Closure under addition and multiplication.
  • The sum is associative and commutative.
  • Multiplication is associative.
  • There exists an additive identity, denoted by 0, such that for all ( a in R, ) ( a + 0 = a. )
  • Every element ( a ) in a ring has an additive inverse ( -a ) such that ( a + (-a) = 0. )
  • Multiplication distributes over addition, that is, ( a, b, c in R, ) ( a times (b + c) = (a times b) + (a times c) ) and ( (b + c) times a = (b times a) + (c times a). )

An ideal is a subset of a ring that absorbs multiplication by elements from the ring and is additive. More technically, a subset ( I ) of a ring ( R ) is an ideal if:

  • For every ( a, b in I, ) the difference ( a - b ) is in ( I. )
  • For every ( r in R ) and ( a in I, ) then ( ra ) and ( ar ) are both in ( I. )

Descending chain position

The descending chain condition (DCC) is a fundamental aspect of Artinian rings. This condition states that every descending chain of ideals in a ring must be eventually constant. In simple terms, if we have a sequence of ideals:

    I_1 supseteq I_2 supseteq I_3 supseteq cdots

There is some integer ( n ) such that ( I_n = I_{n+1} = I_{n+2} = cdots ). This means that after a certain point, all subsequent ideals in the sequence are equal.

Definition of Artinian rings

Taking the above conditions and concepts into account, an Artinian ring is formally defined as a ring that satisfies the descending chain condition on ideals. In other words, any sequence of ideals in a ring that forms a descending chain will eventually become stationary.

Example 1: Finite ring

Any finite ring is an Artinian ring. This is because in a finite set, any strictly descending sequence of elements must be stable, i.e., must become stable because there are a finite number of elements to choose from.

        Consider the ring ( mathbb{Z}/6mathbb{Z} ).
 Ideals in this ring can only be: 1 = ( {0}, ) 2 = ( {0, 2, 4}, ) and 3 = ( {0, 3}. )

Any descending chain involving these ideals must be eventually stationary since no infinite descending sequence is possible. Therefore, ( mathbb{Z}/6mathbb{Z} ) is an Artinian ring.

Characterization of Artinian rings

Beyond the descending chain condition, Artinian rings also exhibit some other interesting properties and characteristics. In particular, Artinian rings are closely related to Noetherian rings, which are defined by the ascending chain condition on ideals. However, while there are similarities between these concepts, each has its own distinct implications and uses in algebra.

Artinian versus Noetherian rings

A ring is called Noetherian if every ascending chain of ideals is stable. Here is a brief comparison:

  • Artinian rings satisfy the descending chain condition on ideals, while Noetherian rings satisfy the ascending chain condition.
  • Every Artinian ring is Noetherian if it is commutative and has unity.
  • Not every Noetherian ring is Artinian.

Example 2: Matrix rings

Consider ( R = M_n(k) ), the ring of ( n times n ) matrices over the field ( k ). This is a classic example of an Artinian ring due to its finite dimension over the field.

Properties and examples

There are some deep implications due to the structural properties of Artinian rings. Let's review some of them with examples:

Property 1: Nilpotent elements and radicals

In an Artinian ring, the Jacobson radical, which is the intersection of all maximal ideals, is nilpotent. Nilpotent means that there is some power of the element that goes to zero.

Example 3: Nilpotent example

In the ring ( R = mathbb{Z}/4mathbb{Z} ), the element 2 is nilpotent since ( 2^2 = 4 equiv 0 (text{mod } 4) ). Since Artinian rings must exhibit this behavior, this is a valid example to demonstrate nilpotent.

Property 2: Simple module

If a ring ( R ) is Artinian, then every module over ( R ) has a combination chain, making modules easier to handle algebraically. A combination chain for a module is a finite chain of submodules where every factor module is simple.

Example 4: Simple module

Consider the Artinian ring ( mathbb{Z}/6mathbb{Z} ), and the module is itself. The combination series is simply ( 0 subset 2mathbb{Z}/6mathbb{Z} subset mathbb{Z}/6mathbb{Z} ), with simple modules ( 2mathbb{Z}/6mathbb{Z} approx mathbb{Z}/2mathbb{Z} ).

Applications of Artinian rings

The concept of Artinian rings is used in various areas of algebra, such as representation theory of algebras and modules, algebraic geometry, and commutative algebra. Here are some areas where Artinian rings play an important role:

1. Algebraic geometry

In algebraic geometry, specific classes of varieties and schemes can be described using Artinian rings. These rings help to understand local properties and special simplifications of algebraic varieties.

2. Representation theory

In representation theory, Artinian rings are often used to describe the endomorphism rings of certain finite-dimensional modules, allowing analysts to study and tackle more complicated representation problems through these simpler structures.

Artinian ring theorem

Several important theorems related to Artinian rings help mathematicians understand and use these structures. Here are some important theorems related to Artinian rings:

Theorem 1: Hopkins–Levitzki theorem

If ( R ) is both Artinian and Noetherian, then all its finitely generated modules are also Artinian and Noetherian. This theorem establishes an essential connection between these two types of rings.

Theorem 2: The Krull–Schmidt theorem

This theorem states that in an Artinian ring, any module decomposes uniquely into a direct sum of indecomposable modules, up to isomorphism and permutation.

Conclusion

Artinian rings serve as an important concept in ring theory, providing a concrete understanding of algebraic structures when considering descending sequences. Their properties and theorems pave the way for further research in algebra, allowing mathematicians to explore both theoretical and practical applications.


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Artinian Rings

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