Glossary

9.1 Definition and Examples of Rings

4 min read•august 16, 2024

Rings are mathematical structures that extend the concept of groups by introducing a second operation. They combine and , following specific rules that govern how these operations interact within the set of elements.

This section introduces the formal definition of rings, outlining their key properties and axioms. We'll explore various examples of rings, from familiar number systems to more abstract structures, laying the groundwork for deeper study in algebra.

Rings and Their Properties

Definition and Structure of Rings

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  • defines algebraic structure consisting of set with two binary operations (addition and multiplication) satisfying specific axioms
  • Addition operation forms abelian group (associative, commutative, identity element 0, additive inverses)
  • Multiplication operation associates and distributes over addition from left and right
  • Rings do not require multiplicative inverses for all non-zero elements
  • Multiplicative identity (1), if present, differs from additive identity (0)
  • Rings classify as unital (with multiplicative identity) or non-unital (without multiplicative identity)

Ring Axioms and Properties

  • Closure under addition and multiplication for all elements in the set
  • of addition: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c) for all a,b,ca, b, c in R
  • Commutativity of addition: a+b=b+aa + b = b + a for all a,ba, b in R
  • Additive identity: a+0=a=0+aa + 0 = a = 0 + a for all aa in R
  • Additive inverses: For each aa in R, there exists −a-a such that a+(−a)=0=(−a)+aa + (-a) = 0 = (-a) + a
  • Associativity of multiplication: (a⋅b)⋅c=a⋅(b⋅c)(a · b) · c = a · (b · c) for all a,b,ca, b, c in R
  • Distributivity of multiplication over addition:
    • Left distributivity: a⋅(b+c)=(a⋅b)+(a⋅c)a · (b + c) = (a · b) + (a · c) for all a,b,ca, b, c in R
    • Right distributivity: (a+b)⋅c=(a⋅c)+(b⋅c)(a + b) · c = (a · c) + (b · c) for all a,b,ca, b, c in R

Examples of Rings

Common Ring Structures

  • () under standard addition and multiplication form with unity
  • with coefficients from a ring (real or complex numbers) create ring under polynomial addition and multiplication
  • Set of n × n matrices over or ring forms ring under matrix addition and multiplication
  • Continuous functions on closed interval [a,b] establish ring under pointwise addition and multiplication of functions
  • Even integers under standard addition and multiplication constitute ring without unity
  • Modular arithmetic systems (integers modulo n, Z/nZ) form rings under addition and multiplication modulo n

Specialized Ring Examples

  • Gaussian integers (complex numbers with integer real and imaginary parts) form commutative ring
  • Quaternions form non-commutative ring with division
  • Ring of formal power series extends polynomial rings to infinite series
  • Boolean rings (elements are idempotent under multiplication) used in logic and computer science
  • Rings of algebraic integers in number fields generalize concept of integers to algebraic number theory

Identifying Rings

Verification Process for Ring Properties

  • Confirm set closure under both addition and multiplication operations
  • Verify addition forms abelian group (associativity, commutativity, additive identity, additive inverses)
  • Check multiplication associativity: (a⋅b)⋅c=a⋅(b⋅c)(a · b) · c = a · (b · c) for all a,b,ca, b, c in set
  • Validate left and right distributive properties:
    • a⋅(b+c)=(a⋅b)+(a⋅c)a · (b + c) = (a · b) + (a · c) for all a,b,ca, b, c in set
    • (a+b)⋅c=(a⋅c)+(b⋅c)(a + b) · c = (a · c) + (b · c) for all a,b,ca, b, c in set
  • Ensure additive and multiplicative identities (if present) differ
  • Recognize multiplicative identity not necessary for ring, but defines unital ring if present

Common Pitfalls and Considerations

  • Watch for potential counterexamples violating ring axioms
  • Pay attention to closure property, often overlooked in non-standard structures
  • Verify distributivity carefully, common source of errors in ring identification
  • Consider special cases or boundary conditions that might violate ring properties
  • Check for existence of zero divisors (non-zero elements whose product is zero)
  • Examine commutativity of multiplication, not required for general rings

Commutative vs Non-commutative Rings

Characteristics of Commutative Rings

  • Commutative rings satisfy a⋅b=b⋅aa · b = b · a for all elements aa and bb
  • Integers (Z) and polynomial rings over commutative rings exemplify commutative rings
  • Commutative rings often possess simpler algebraic properties
  • Close relationship to number theory and algebraic geometry
  • Examples: real numbers (R), complex numbers (C), rational numbers (Q)
  • Commutative ring theory connects to theory and module theory

Properties of Non-commutative Rings

  • Non-commutative rings have at least one pair of elements where a⋅b≠b⋅aa · b ≠ b · a
  • Ring of n × n matrices (n > 1) over field represents classic non-commutative ring
  • Applications in quantum mechanics, representation theory, and advanced mathematics
  • Examples: quaternions, octonions, matrix rings
  • Center of ring (elements commuting with all others) always forms commutative
  • Non-commutative ring theory relates to operator algebras and non-commutative geometry

Key Terms to Review (18)

Addition: Addition is a fundamental binary operation that combines two elements to produce a third element within a mathematical structure, such as a ring or field. It forms the basis for understanding how numbers and other algebraic structures behave when combined, which is crucial in defining properties like associativity, commutativity, and identity elements in rings and fields.
Associativity: Associativity is a fundamental property in mathematics that describes how the grouping of elements affects the result of an operation. Specifically, an operation is associative if changing the grouping of the operands does not change the outcome; mathematically, this means for an operation * and elements a, b, and c, the equation (a * b) * c = a * (b * c) holds true. This property is crucial in various algebraic structures as it ensures consistency and predictability when performing operations, especially in systems like groups, direct products, and rings.
Commutative ring: A commutative ring is an algebraic structure consisting of a set equipped with two binary operations: addition and multiplication, satisfying certain axioms. In a commutative ring, the multiplication operation is commutative, meaning that the order of multiplication does not affect the result. This structure plays a vital role in understanding both abstract algebra and various mathematical concepts, especially in relation to ideals and quotient rings.
David Hilbert: David Hilbert was a German mathematician known for his foundational contributions to mathematics, particularly in areas like algebra, number theory, and geometry. His work laid essential groundwork for many modern mathematical theories and concepts, influencing various fields including rings, integral domains, and representation theory.
Distributive Property: The distributive property is a fundamental algebraic principle that states that for any real numbers a, b, and c, the equation a(b + c) = ab + ac holds true. This property allows for the multiplication of a single term by a sum or difference within parentheses, distributing the term across each component inside. It's crucial for simplifying expressions and solving equations in various mathematical contexts, including ring theory where it applies to the operations of addition and multiplication.
Emmy Noether: Emmy Noether was a pioneering mathematician known for her groundbreaking work in abstract algebra and theoretical physics, particularly in the areas of ring theory and the development of what is now known as Noetherian rings. Her contributions established deep connections between algebra and geometry, leading to fundamental insights into structures such as integral domains and fields, as well as the characteristic of rings. Noether's work laid the foundation for modern algebra, influencing various mathematical disciplines and providing essential tools for future discoveries.
Field: A field is a set equipped with two operations, typically called addition and multiplication, satisfying certain properties that allow for the manipulation of its elements. These properties include commutativity, associativity, distributivity, the existence of additive and multiplicative identities, and the existence of inverses for every non-zero element. Fields are essential in various mathematical structures, including rings and integral domains, as they provide a way to perform arithmetic and solve equations more broadly.
Ideal: An ideal is a special subset of a ring that absorbs multiplication by elements of the ring, serving as a building block for constructing new rings and studying their properties. Ideals play a crucial role in ring theory, allowing for the development of quotient rings and providing insight into the structure of rings. They can also be linked to important concepts such as homomorphisms and factorization within algebraic systems.
Integers: Integers are the set of whole numbers that include all positive whole numbers, negative whole numbers, and zero. They form a fundamental concept in mathematics and are important in various mathematical structures, including rings. Integers allow for operations like addition, subtraction, and multiplication to be performed without leaving the set, making them integral to number theory and algebraic structures.
Integral domain: An integral domain is a type of commutative ring with unity that has no zero divisors and is an important structure in abstract algebra. It extends the concept of integers, allowing for polynomial and algebraic structures to operate without encountering non-invertible elements. This property is crucial as it supports unique factorization, making it a foundational aspect in number theory and algebraic geometry.
Multiplication: Multiplication is a binary operation that combines two elements to produce a third element, following specific rules depending on the algebraic structure involved. In the context of rings, multiplication is an essential operation that must satisfy certain properties such as associativity and distributivity. In integral domains and fields, multiplication must also adhere to additional criteria, such as the existence of multiplicative inverses in fields, which further influences their structural characteristics.
Polynomials: Polynomials are mathematical expressions consisting of variables, coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication. They serve as fundamental building blocks in algebra and can represent a variety of functions, making them essential in various mathematical fields, including ring theory. Polynomials can be classified by their degree, which is determined by the highest exponent present in the expression, and can also be evaluated and manipulated within structures known as rings.
R: 'r' is commonly used to denote a ring in abstract algebra, which is a set equipped with two operations satisfying certain axioms. In the context of rings, 'r' represents not only the set itself but also serves as a symbol for elements within that set. The structure of 'r' is fundamental as it showcases properties such as closure under addition and multiplication, the existence of an additive identity, and potentially the presence of a multiplicative identity, depending on whether we are dealing with a ring with unity or not.
Ring: A ring is an algebraic structure consisting of a set equipped with two binary operations, typically referred to as addition and multiplication, where the set is closed under these operations and satisfies specific properties like associativity, distributivity, and the existence of an additive identity. This concept connects to various mathematical structures and underpins the study of more complex systems like integral domains and fields, as well as applications in areas such as Galois theory.
Ring homomorphism: A ring homomorphism is a function between two rings that preserves the operations of addition and multiplication. Specifically, if \(f: R \to S\) is a ring homomorphism from ring \(R\) to ring \(S\), then for all elements \(a, b \in R\), it holds that \(f(a + b) = f(a) + f(b)\) and \(f(a \cdot b) = f(a) \cdot f(b)\). This concept is essential in understanding the structure of rings and their interrelations, as it allows for the transfer of properties between different rings, which is critical when exploring ideals and quotient rings.
Subring: A subring is a subset of a ring that is itself a ring under the same operations of addition and multiplication as the original ring. For a subset to qualify as a subring, it must contain the additive identity, be closed under addition and multiplication, and include the additive inverses of its elements. This concept is crucial in understanding how rings can be structured and analyzed.
Unit element: A unit element, often referred to as a multiplicative identity, is an element in a ring such that when it is multiplied by any element in the ring, it leaves that element unchanged. This means that if '1' is the unit element in a ring, for any element 'a' in the ring, the equation '1 * a = a' holds true. The presence of a unit element is crucial for the structure of a ring, helping to define the concept of invertibility and forming the basis for understanding fields and division rings.
Z: In the context of rings, 'z' typically refers to the set of integers, denoted as $$ ext{Z}$$. This set is fundamental in abstract algebra because it forms a ring under the standard operations of addition and multiplication. The ring of integers is an example of an integral domain, showcasing essential properties such as commutativity and the existence of additive and multiplicative identities.
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