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What is a Set, a Group, a Ring, and a Field?

Exploring Fundamental Abstract Algebra



Set

A set in mathematics is a collection of distinct objects, considered as an object in its own right. These objects are called the elements or members of the set. (Set - Wolfram MathWorld)


Group

A group is a set equipped with an operation that combines any two elements to form a third element in such a way that it is associative, has an identity element, and every element has an inverse. (Group - Wolfram MathWorld)


Abelian Group

An abelian group is a group in which the group operation is commutative, meaning that the order in which two elements are combined does not affect the result. In other words, for any two elements a and b in the group, ab=ba. (Abelian Group - Wolfram MathWorld)


Ring

A ring is a set equipped with two binary operations - (usually referred to as addition and multiplication, or by the symbols + and x) - where addition forms a group and multiplication is associative, with the distributive property holding over addition. (Ring - Wolfram MathWorld)


FielD

A field is a set equipped with two binary operations - (usually referred to as addition and multiplication, or by the symbols + and x) - that satisfy the following properties:


Fields serve as a fundamental framework for various areas of algebra, including number theory, algebraic geometry, and algebraic topology, providing a structure for defining and solving equations and analysing polynomial functions.


Building on the definition of a ring, above, a field is a ring in which every non-zero element has a multiplicative inverse, there exists a multiplicative identity, and multiplication is commutative.


Abstract Algebra

Fundamental abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. These structures form the backbone of various mathematical disciplines, including number theory, topology, and algebraic geometry. Abstract algebra explores the deep relationships between these structures through their axioms and properties, such as commutativity, associativity, and the presence of inverses. This field is crucial for both theoretical and applied mathematics, providing the tools necessary to solve polynomial equations, understand symmetry, and develop modern cryptography systems.


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