7.5 Radical theory for non-associative rings

Radical theory for non-associative rings extends classical concepts to algebras without associativity. It provides tools to analyze the structure of diverse algebraic systems by identifying and isolating their "bad" or "pathological" parts.

This theory explores various types of radicals, each capturing different algebraic properties. Understanding these radicals offers insights into the behavior of non-associative rings and allows for their classification based on radical properties.

Definition of radical theory

• Radical theory provides a framework for studying algebraic structures by identifying and isolating their "bad" or "pathological" parts
• In non-associative algebra, radical theory extends classical concepts to rings without associativity, offering insights into their structure and properties

Concepts of radicals

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• Radicals identify maximal ideals with specific properties within algebraic structures
• Generalize the notion of nilpotent elements in associative rings to non-associative contexts
• Serve as a tool for decomposing rings into "good" (semisimple) and "bad" (radical) parts
• Allow for classification of rings based on their radical properties

Historical development

• Originated in the early 20th century with the work of Wedderburn on associative algebras
• Extended to non-associative algebras by Albert and Zorn in the 1930s
• Kurosh and Amitsur formalized the general theory of radicals in the 1950s
• Recent developments focus on categorical and topological approaches to radical theory

Types of radicals

• Radical theory in non-associative algebra explores various types of radicals, each capturing different algebraic properties
• Understanding these radicals provides insights into the structure and behavior of non-associative rings

Jacobson radical

• Generalizes the concept of Jacobson radical from associative rings to non-associative settings
• Defined as the intersection of all maximal modular right ideals of a ring
• Characterizes elements that act nilpotently on all simple modules
• Plays a crucial role in the structure theory of non-associative rings
  • Helps identify the semisimple part of a ring

Brown-McCoy radical

• Extends the notion of Brown-McCoy radical to non-associative rings
• Defined as the intersection of all maximal two-sided ideals of a ring
• Captures the largest ideal contained in all maximal ideals
• Useful in studying the structure of non-associative rings without identity elements
  • Provides insights into the relationship between one-sided and two-sided ideals

Baer radical

• Generalizes the concept of nil radicals to non-associative contexts
• Defined as the intersection of all prime ideals of a ring
• Characterizes the nilpotent elements in non-associative rings
• Important in studying the nilpotent structure of non-associative algebras
  • Helps identify rings with non-trivial nilpotent elements

Properties of radicals

• Radicals in non-associative algebra exhibit specific properties that allow for their systematic study and application
• These properties form the foundation for developing a comprehensive theory of radicals in non-associative contexts

Hereditary properties

• Radicals preserve their properties when passing to subrings or ideals
• Hereditary radicals remain unchanged under the formation of subrings
• Important for studying the behavior of radicals in quotient rings and extensions
• Examples of hereditary radicals include
  • Baer radical in non-associative rings
  • Locally nilpotent radical in Lie algebras

Homomorphic properties

• Radicals behave consistently under ring homomorphisms
• Image of a radical under a surjective homomorphism remains a radical
• Allows for the study of radicals in homomorphic images and quotient rings
• Crucial for developing structure theorems in non-associative algebra
  • Enables the analysis of radicals in direct products and direct sums of rings

Idempotent properties

• Some radicals exhibit idempotent behavior $R(R(A)) = R(A)$
• Idempotent radicals simplify the study of iterated radical constructions
• Important for understanding the stability of radical classes
• Examples of idempotent radicals in non-associative contexts
  • Jacobson radical in alternative rings
  • Nil radical in Jordan algebras

Radical classes

• Radical classes provide a systematic way to study and classify non-associative rings based on their radical properties
• These classes form the foundation for developing a comprehensive theory of radicals in non-associative algebra

Kurosh-Amitsur radical classes

• Generalize the concept of radical classes from associative rings to non-associative settings
• Defined by a set of axioms that ensure closure under certain operations
• Include important radical classes such as
  • Jacobson radical class
  • Baer radical class
• Allow for the systematic study of radicals across different types of non-associative rings
  • Provide a unified framework for analyzing radicals in Lie algebras, Jordan algebras, and alternative rings

Special radical classes

• Capture specific algebraic properties in non-associative contexts
• Include radical classes tailored to particular types of non-associative rings
• Examples of special radical classes
  • Solvable radical class in Lie algebras
  • Penico radical class in alternative rings
• Provide insights into the structure and behavior of specific families of non-associative rings
  • Allow for the development of specialized decomposition theorems and structure results

Semisimple classes

• Semisimple classes in non-associative algebra complement radical classes, providing a dual perspective on ring structure
• These classes play a crucial role in understanding the "well-behaved" parts of non-associative rings

Definition and examples

• Semisimple classes consist of rings with trivial radical (zero radical)
• Characterized by the absence of certain "bad" properties associated with radicals
• Examples of semisimple classes in non-associative contexts
  • Simple Lie algebras
  • Finite-dimensional semisimple alternative algebras
• Provide important structural information about non-associative rings
  • Help identify rings with "nice" algebraic properties (simple, completely reducible)

Relationship to radicals

• Semisimple classes form a complementary pair with radical classes
• Every ring can be decomposed into its radical part and semisimple part
• Semisimple quotient $A/R(A)$ belongs to the corresponding semisimple class
• Understanding this relationship allows for a complete structural analysis of non-associative rings
  • Enables the development of structure theorems and classification results

Radical theory for non-associative rings

• Radical theory for non-associative rings extends classical concepts to algebras without associativity
• This extension provides powerful tools for analyzing the structure and properties of diverse algebraic systems

Challenges in non-associative context

• Lack of associativity complicates the definition and properties of ideals
• Traditional radical constructions may not behave as expected in non-associative settings
• Need to account for different types of non-associativity (alternative, Jordan, Lie)
• Requires careful adaptation of proofs and techniques from associative theory
  • Demands new approaches to handle the peculiarities of non-associative multiplication

Adaptations of associative theory

• Modify definitions of radicals to accommodate non-associative multiplication
• Develop new techniques for constructing and analyzing radicals in specific non-associative contexts
• Introduce specialized radicals tailored to particular classes of non-associative rings
• Extend important theorems and results from associative theory to non-associative settings
  • Generalize the Wedderburn-Artin theorem for alternative algebras

Applications of radical theory

• Radical theory in non-associative algebra finds applications in various areas of mathematics and related fields
• These applications demonstrate the power and versatility of radical theory as a tool for studying algebraic structures

Structure theory

• Radical decompositions provide insights into the internal structure of non-associative rings
• Allows for classification of rings based on their radical and semisimple components
• Helps identify simple and semisimple subalgebras within complex non-associative structures
• Applications in the study of
  • Lie algebras and their representations
  • Jordan algebras and their relation to quantum mechanics

Representation theory

• Radicals play a crucial role in understanding the representation theory of non-associative algebras
• Jacobson radical helps identify faithful and irreducible representations
• Allows for the construction of composition series for modules over non-associative rings
• Applications in
  • Studying representations of Lie algebras in particle physics
  • Analyzing Jordan algebra representations in quantum information theory

Ideal theory

• Radical theory provides tools for studying the ideal structure of non-associative rings
• Helps identify and classify different types of ideals (prime, maximal, minimal)
• Allows for the development of primary decomposition theorems in non-associative contexts
• Applications in
  • Studying the ideal structure of enveloping algebras of Lie algebras
  • Analyzing ideals in octonion algebras and exceptional Jordan algebras

Radical-semisimple decomposition

• Radical-semisimple decomposition provides a fundamental tool for understanding the structure of non-associative rings
• This decomposition allows for the separation of "well-behaved" and "pathological" parts of algebraic structures

Theorem and proof

• States that every non-associative ring A can be decomposed as $A = R(A) \oplus S$
• R(A) represents the radical of A, and S is a semisimple complement
• Proof involves constructing the semisimple complement using idempotent elements
• Key steps in the proof
  • Show that R(A) is the largest ideal with trivial intersection with any semisimple subalgebra
  • Construct a maximal semisimple subalgebra S
  • Demonstrate that A = R(A) + S and R(A) ∩ S = 0

Examples in non-associative rings

• Radical-semisimple decomposition of a finite-dimensional Lie algebra
  • Decompose into solvable radical and semisimple Levi subalgebra
• Peirce decomposition of a Jordan algebra
  • Separate idempotent and nilpotent components
• Decomposition of an alternative algebra into its Jacobson radical and semisimple part
• Applications in studying the structure of
  • Malcev algebras and their representations
  • Non-associative division algebras (octonions)

Radical operations

• Radical operations in non-associative algebra provide tools for combining and manipulating radicals
• These operations allow for a deeper understanding of the relationships between different radical classes

Sum of radicals

• Defines the sum of two radicals as the smallest radical containing both
• Allows for the construction of new radicals from existing ones
• Important in studying the lattice structure of radical classes
• Examples of radical sums in non-associative contexts
  • Sum of nil radical and locally nilpotent radical in Lie algebras
  • Combining Jacobson and Brown-McCoy radicals in alternative rings

Intersection of radicals

• Defines the intersection of radicals as the largest radical contained in both
• Provides a way to identify common properties of different radical classes
• Useful in studying the relationships between various radical constructions
• Examples of radical intersections in non-associative algebra
  • Intersection of Baer and Jacobson radicals in Jordan algebras
  • Finding common elements of solvable and nilpotent radicals in Lie algebras

Radical theory vs other theories

• Radical theory in non-associative algebra interacts with and complements other algebraic theories
• Understanding these relationships provides a broader perspective on the structure of non-associative rings

Radical theory vs torsion theory

• Radical theory focuses on identifying "bad" subrings, while torsion theory studies divisibility properties
• Both theories provide tools for decomposing rings into well-behaved and pathological parts
• Similarities
  • Both use closure operations to define classes of rings
  • Both allow for the study of quotient structures
• Differences
  • Radical theory applies to a wider range of algebraic structures
  • Torsion theory is more closely tied to module theory

Radical theory vs localization theory

• Radical theory identifies and isolates "bad" subrings, while localization theory focuses on inverting elements
• Both theories provide ways to simplify the study of algebraic structures
• Connections
  • Localization can sometimes eliminate radicals (localizing at semiprime ideals)
  • Radical properties can influence localization behavior
• Differences
  • Radical theory applies to a broader class of non-associative rings
  • Localization theory requires additional conditions (existence of multiplicative sets) in non-associative settings

Advanced topics

• Advanced topics in radical theory for non-associative algebra explore new directions and generalizations
• These areas of research push the boundaries of radical theory and connect it to other branches of mathematics

Topological radical theory

• Extends radical theory to topological non-associative rings and algebras
• Introduces concepts of closed and dense radicals in topological settings
• Studies the interplay between algebraic and topological properties of radicals
• Applications in
  • Analyzing radicals in Banach-Lie algebras
  • Studying topological Jordan algebras and their representations

Categorical approach to radicals

• Develops a general framework for radical theory using category theory
• Allows for the study of radicals in a wide range of algebraic structures
• Introduces concepts like radical functors and torsion theories in non-associative categories
• Provides new insights into
  • Universal properties of radical constructions
  • Connections between different types of non-associative algebras through their radical structures