What is a Set, a Group, a Ring, and a Field?
Exploring Fundamental Abstract Algebra
Set
A set 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 (which also must belong to the set) in such a way that it is associative, has an identity element, and every element has an inverse with respect to the given operation. (Group - Wolfram MathWorld)
Associative
An operation is called "associative" if changing the grouping of the operands does not change the result of the operation. This means that if the operation is denoted by "* ", then for any elements a, b, and c, the equation:
(a * b) * c = a * (b * c)
always holds.
Identity Element
An identity element is an element in a set that, when combined with any other element of the set through a specific binary operation, leaves the other element unchanged. This means for a binary operation "* " defined on a set S, an element e in S is an identity element if, for every element a in S :
e * a = a and a * e = a
The identity element does not alter other elements under the operation. Common examples include 0 for addition and 1 for multiplication.
Inverse
An inverse under a particular binary operation is an element that, when combined with another element, results in the identity element of that operation. Specifically, for an operation "* " defined on a set, if e is the identity element, then for any element a in the set, an element b - often denoted as:
a^{-1}
is called the inverse of a if:
a * b = e and b * a = e
The inverse reverses the effect of combining with the original element under the given operation. Common examples include the additive inverse -a for addition, where:
a + (-a) = 0
and the multiplicative inverse:
a^{-1}
for multiplication, where:
a x a^{-1} = 1
Abelian Group
An abelian group is a group in which the group operation is also commutative (as well as associative), 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, a * b=b * a. (Abelian Group - Wolfram MathWorld)
Commutative
An operation is "commutative" if the order in which two elements are combined does not affect the result. This means that for a binary operation "* " defined on a set, the operation is commutative if for any elements a and b in the set:
a * b = b * a
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)
Put another way (without defining a ring in terms of a group) - a ring is a set equipped with two binary operations that generalise the arithmetic operations of addition and multiplication:
Closure under Addition and Multiplication: For all a, b ∈ R, both a + b and a * b are also in R
Associativity of Addition and Multiplication: For all a, b ∈ R, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)
Commutativity of Addition: For all a, b ∈ R, a + b = b + a
Existence of Additive Identity: There exists an element 0 ∈ R such that a + 0 = a for all a ∈ R
Existence of Additive Inverses: For every a ∈ R, there exists an element -a ∈ R such that a + (-a) = 0
Distributive Laws: For all a, b, c ∈ R, a * (b + c) = a * b + a * c and (b + c) * a = b * a + c * a
Multiplicative Identity (not required in all definitions): In some definitions, particularly for rings that are also termed unital or unitary, there exists an element 1 ∈ R (distinct from 0 ) such that a * 1 = a and 1 * a = 1 for all a ∈ R
Commutativity of Multiplication (not required in all definitions): In a commutative ring, for all a, b ∈ R, a * b = b * a . However, this property is not required for a set to be a ring. Without this property, the set is called a "non-commutative ring"
FielD
A field is a set equipped with two binary operations - (usually referred to as addition and multiplication, or by the symbols + and * ) - that satisfy the following properties:
Closure: For any two elements a and b in the field, both a + b and a * b are also in the field. (Wolfram MathWorld)
Associativity: Addition and multiplication are associative; that is, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c) for any elements a, b, and c in the field. (Wolfram MathWorld)
Commutativity: Both addition and multiplication are commutative, meaning a + b = b + a and a * b = b * a. (Wolfram MathWorld)
Identity Elements: There exists an additive identity (usually denoted as 0 and a multiplicative identity (usually denoted as 1 ) in the field, such that a + 0 = a and a * 1 = a for any element a. (Wolfram MathWorld)
Additive Inverses: For every element a, there exists an element -a such that a + (-a) = 0. (Wolfram MathWorld)
Multiplicative Inverses: For every non-zero element a, there exists an element a^{-1} such that a x a^{-1} = 1, except for the additive identity 0. (Wolfram MathWorld)
Distributivity: Multiplication distributes over addition, i.e., a * (b + c) = (a * b) + (a * c). (Wolfram MathWorld)
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.
