Algebraic Structures

Mathematics > Abstract Algebra > Algebraic Structures

Abstract Algebra is a branch of mathematics that studies algebraic systems in a broad, abstract manner rather than focusing on numbers and their properties as in elementary algebra. Within this expansive field, algebraic structures form a foundational concept.

Algebraic Structures: An Overview

An algebraic structure is a set equipped with one or more operations that satisfy a given set of axioms. These structures generalize both arithmetical operations and the rules of algebra we first learn in elementary mathematics. The axioms help define how the operations interact with the elements of the set, and different algebraic structures have different roles and properties based on these definitions.

Essential Types and Examples

1. Groups: A group is an algebraic structure consisting of a set \( G \) and a single binary operation (often denoted as a multiplication or addition) that combines any two elements to form another element of the same set. The key properties that define a group are closure, associativity, the existence of an identity element, and the existence of an inverse element for every element in the set. \[ \{G, \cdot \}, \quad \text{such that for all } a, b, c \in G: \]
   • Closure: \( a \cdot b \in G \)
   • Associativity: \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \)
   • Identity element: There exists an \( e \in G \) such that \( e \cdot a = a \cdot e = a \)
   • Inverse element: For each \( a \in G \), there exists an \( a^{-1} \in G \) such that \( a \cdot a^{-1} = a^{-1} \cdot a = e \)
2. Rings: A ring is an algebraic structure comprised of a set \( R \) equipped with two binary operations typically referred to as addition and multiplication. These operations make \( R \) an abelian group with respect to addition, and associative with respect to multiplication. Further, multiplication distributes over addition. \[ \{R, +, \cdot \}, \quad \text{such that for all } a, b, c \in R: \]
   • \( R \) is an abelian group under addition:
     • Associativity: \( a + (b + c) = (a + b) + c \)
     • Commutativity: \( a + b = b + a \)
     • Identity element: \( 0 \in R \quad \text{with} \quad 0 + a = a + 0 = a \)
     • Inverse elements: \( a \in R \) has a \( -a \in R \) such that \( a + (-a) = 0 \)
   • Multiplication is associative: \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \)
   • Distributivity of multiplication over addition:
     • Left distributivity: \( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \)
     • Right distributivity: \( (a + b) \cdot c = (a \cdot c) + (b \cdot c) \)
3. Fields: A field is an algebraic structure that is essentially a ring with additional properties, allowing division by non-zero elements. A field includes a set \( F \) equipped with two operations, addition and multiplication, satisfying the properties of both a commutative group under addition and a commutative monoid under multiplication, with the additional feature that every non-zero element has a multiplicative inverse. \[ \{F, +, \cdot \}, \quad \text{such that for all } a, b, c \in F: \]
   • \( F \) is an abelian group under addition.
   • \( F \) is a commutative monoid under multiplication:
     • Associativity: \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \)
     • Commutativity: \( a \cdot b = b \cdot a \)
     • Identity element: \( 1 \in F \) with \( 1 \cdot a = a \cdot 1 = a \)
   • Every non-zero element \( a \in F \) has a multiplicative inverse \( a^{-1} \) such that \( a \cdot a^{-1} = a^{-1} \cdot a = 1 \)

These are just a few examples of the many possible algebraic structures in abstract algebra, each providing a framework that allows for the generalization and analysis of concepts initially developed in the context of numbers and arithmetic. The study of algebraic structures not only deepens our understanding of mathematical theory but also finds extensive applications in areas such as cryptography, coding theory, and the theory of computation.