9.1 Definition and Examples of Rings

Rings are mathematical structures that extend the concept of groups by introducing a second operation. They combine addition and multiplication, 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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• Ring defines algebraic structure consisting of set R 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
• Associativity of addition: $(a + b) + c = a + (b + c)$ for all $a, b, c$ in R
• Commutativity of addition: $a + b = b + a$ for all $a, b$ in R
• Additive identity: $a + 0 = a = 0 + a$ for all $a$ in R
• Additive inverses: For each $a$ in R, there exists $-a$ such that $a + (-a) = 0 = (-a) + a$
• Associativity of multiplication: $(a · b) · c = a · (b · c)$ for all $a, b, c$ in R
• Distributivity of multiplication over addition:
  • Left distributivity: $a · (b + c) = (a · b) + (a · c)$ for all $a, b, c$ in R
  • Right distributivity: $(a + b) · c = (a · c) + (b · c)$ for all $a, b, c$ in R

Examples of Rings

Common Ring Structures

• Integers (Z) under standard addition and multiplication form commutative ring with unity
• Polynomials with coefficients from a ring (real or complex numbers) create ring under polynomial addition and multiplication
• Set of n × n matrices over field 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: $(a · b) · c = a · (b · c)$ for all $a, b, c$ in set
• Validate left and right distributive properties:
  • $a · (b + c) = (a · b) + (a · c)$ for all $a, b, c$ in set
  • $(a + b) · c = (a · c) + (b · c)$ for all $a, b, c$ 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 $a · b = b · a$ for all elements $a$ and $b$
• 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 ideal theory and module theory

Properties of Non-commutative Rings

• Non-commutative rings have at least one pair of elements where $a · b ≠ b · a$
• 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 subring
• Non-commutative ring theory relates to operator algebras and non-commutative geometry