Glossary

1.1 Definition and examples of commutative rings

2 min readโ€ขjuly 25, 2024

Commutative rings are mathematical systems with two operations: addition and multiplication. They follow specific rules that govern how elements interact, like associativity and commutativity. These structures form the foundation for more advanced algebraic concepts.

Examples of commutative rings include integers, polynomials, and . Each type has unique properties and applications in different areas of mathematics. Understanding these examples helps illustrate the broader principles of commutative algebra.

Fundamental Concepts of Commutative Rings

Definition of commutative rings

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  • structure forms mathematical system with two binary operations (addition and multiplication) closed under both operations
  • Axioms of a commutative ring govern behavior of elements:
    • Associativity of addition allows grouping elements flexibly (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)
    • Commutativity of addition permits rearranging terms a+b=b+aa + b = b + a
    • Additive identity element (usually 0) leaves other elements unchanged a+0=aa + 0 = a
    • Additive inverse element negates original element a+(โˆ’a)=0a + (-a) = 0
    • Associativity of multiplication enables flexible grouping (ab)c=a(bc)(ab)c = a(bc)
    • Commutativity of multiplication allows rearranging factors ab=baab = ba
    • element (usually 1) preserves other elements aโ‹…1=aa \cdot 1 = a
    • Distributive property connects addition and multiplication a(b+c)=ab+aca(b + c) = ab + ac
  • Notation (R,+,โ‹…)(R, +, \cdot) or RR concisely represents commutative ring structure

Examples of commutative rings

  • Z\mathbb{Z} encompasses whole numbers with familiar addition and multiplication
  • Polynomial rings F[x]F[x] over a FF consist of polynomial expressions with coefficients from FF
  • Gaussian integers Z[i]\mathbb{Z}[i] extend complex numbers to include integer real and imaginary parts
  • Ring of in a number field generalizes integers to algebraic number theory
  • Quotient rings Z/nZ\mathbb{Z}/n\mathbb{Z} represent integers modulo nn, useful in modular arithmetic
  • Matrix rings Mn(R)M_n(R) over a commutative ring RR only commutative for 1ร—11 \times 1 matrices

Identity and inverse elements

  • Multiplicative identity (unity) uniquely satisfies 1โ‹…a=aโ‹…1=a1 \cdot a = a \cdot 1 = a for all aa in RR
  • aโˆ’1a^{-1} fulfills aโ‹…aโˆ’1=aโˆ’1โ‹…a=1a \cdot a^{-1} = a^{-1} \cdot a = 1 when it exists
  • form group under multiplication, consisting of elements with multiplicative inverses
  • Integral domains exclude , preventing ab=0ab = 0 for non-zero aa and bb
  • Fields extend commutative rings by ensuring every non-zero element has a multiplicative inverse (rational numbers, real numbers)

Commutative vs non-commutative rings

  • Commutative rings satisfy ab=baab = ba for all elements, simplifying algebraic manipulations
  • rings have abโ‰ baab \neq ba for some elements, adding complexity (matrices, quaternions)
  • Commutativity enables more general theorems and algebraic structures
  • contains elements commuting with all others, equaling entire ring for commutative rings

Key Terms to Review (15)

Algebraic Integers: Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They generalize the concept of integers to more abstract number systems, playing a crucial role in understanding the structure of rings. Algebraic integers help define various important algebraic structures, particularly within the realm of commutative rings where they can exhibit unique properties and relationships with other elements.
Center of a Ring: The center of a ring is the set of elements that commute with every element in the ring. In other words, if you take an element from the center and multiply it by any element in the ring, the result is the same regardless of the order of multiplication. This concept is crucial for understanding the structure and properties of rings, especially in distinguishing between commutative and non-commutative rings, and it plays a significant role in examining ideal theory and module theory.
Commutative Ring: A commutative ring is an algebraic structure consisting of a set equipped with two operations, addition and multiplication, satisfying certain properties that include associativity, distributivity, and commutativity for multiplication. In this structure, the multiplication operation is commutative, meaning that the order in which two elements are multiplied does not affect the product. This concept is foundational in mathematics as it relates to subrings, ideals, and how rings can be transformed through homomorphisms and isomorphisms.
Field: A field is a set equipped with two operations, addition and multiplication, that satisfy certain properties, allowing for the division of non-zero elements. Fields play a crucial role in algebra since they provide a structure where every non-zero element has a multiplicative inverse, making them essential in understanding commutative rings and integral domains. The properties of fields enable operations such as finding quotients and establishing isomorphisms between algebraic structures.
Gaussian Integers: Gaussian integers are complex numbers of the form $$a + bi$$, where both $$a$$ and $$b$$ are integers, and $$i$$ is the imaginary unit with the property that $$i^2 = -1$$. This set of numbers forms a ring, which means they can be added and multiplied together following specific rules. Gaussian integers expand the concept of integers into the complex plane, offering new ways to solve equations and explore properties within algebra.
Integer Ring: An integer ring is a specific type of commutative ring consisting of the set of integers, denoted by $$ ext{Z}$$, where addition and multiplication are defined as usual. This ring possesses the properties of closure under addition and multiplication, has an additive identity (0), and a multiplicative identity (1), making it a fundamental example of a commutative ring that illustrates essential algebraic concepts.
Integral Domain: An integral domain is a type of commutative ring with no zero divisors, meaning that the product of any two non-zero elements is always non-zero. This property ensures that integral domains have certain arithmetic characteristics similar to those of integers, making them foundational in the study of algebraic structures.
Matrix Ring: A matrix ring is a set of matrices of a fixed size over a given ring that forms a ring itself under standard matrix addition and multiplication. In the context of commutative algebra, matrix rings illustrate how structures can behave differently when we move from scalar rings to matrix rings, highlighting key properties like non-commutativity in general cases.
Multiplicative Identity: The multiplicative identity is the element in a ring that, when multiplied by any other element in that ring, leaves the other element unchanged. This property is crucial because it establishes a fundamental building block for the structure of commutative rings, ensuring that every element can interact with this identity without altering its value. The multiplicative identity is typically represented by the number 1, which plays an essential role in the arithmetic operations within these rings.
Multiplicative Inverse: The multiplicative inverse of a number is another number which, when multiplied together with the original number, yields the identity element for multiplication, which is 1. In the context of rings, every non-zero element must have a multiplicative inverse in a field, while in a general ring, not every element is guaranteed to have one. Understanding multiplicative inverses helps to grasp the structure and properties of commutative rings.
Non-commutative: Non-commutative refers to mathematical structures where the order of operations matters; specifically, in non-commutative rings, multiplication does not satisfy the commutative property. This means that for two elements a and b in such a ring, it is possible that a * b is not equal to b * a. Understanding non-commutativity is essential for recognizing the differences between various algebraic structures, including how they interact with their elements and the kinds of problems they can solve.
Polynomial Ring: A polynomial ring is a mathematical structure formed by the set of polynomials with coefficients from a given commutative ring, together with the operations of polynomial addition and multiplication. This construction allows us to work with polynomials as elements of a ring, preserving the essential properties like associativity, commutativity, and distributivity. Polynomial rings are crucial in algebra, as they generalize the notion of polynomials over fields and provide a framework for solving polynomial equations in a broader context.
Quotient Ring: A quotient ring is a type of ring formed by taking a commutative ring and dividing it by one of its ideals. It provides a way to construct new rings from existing ones, revealing important algebraic structures and properties. Quotient rings are crucial in understanding the relationships between rings and their ideals, especially in the context of isomorphism theorems and the characterization of prime and maximal ideals.
Units: In the context of commutative rings, a unit is an element that has a multiplicative inverse within the ring. This means that for a unit 'u', there exists another element 'v' in the ring such that the product of 'u' and 'v' equals the multiplicative identity, typically denoted as 1. The presence of units is crucial as they help to form the structure of the ring and determine its properties, especially in relation to invertible elements and factorization.
Zero Divisors: Zero divisors are non-zero elements in a commutative ring such that their product equals zero. This means if you multiply two zero divisors, the result is the additive identity, or zero, which reveals an interesting aspect of the ring's structure. The presence of zero divisors indicates that the ring is not an integral domain, a more restricted type of ring where such elements cannot exist.
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