Integral domain

Definition

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.

5 Must Know Facts For Your Next Test

1. Every field is an integral domain because fields do not have zero divisors.
2. An integral domain must have at least two distinct elements: 0 (the additive identity) and 1 (the multiplicative identity).
3. The polynomial ring over an integral domain is also an integral domain, which means that polynomials can be factored uniquely.
4. Examples of integral domains include the set of integers, polynomial rings over fields, and certain subsets of rational numbers.
5. The characteristic of an integral domain can be either zero or a prime number, impacting the structure of the ring.

Review Questions

• What distinguishes an integral domain from other types of rings?
  • An integral domain is distinguished by its lack of zero divisors and its requirement for a unity element. Unlike general rings that may have zero divisors (elements 'a' and 'b' where 'a * b = 0' but neither 'a' nor 'b' is zero), an integral domain ensures that if the product of two elements is zero, at least one of those elements must be zero. This property enables unique factorization and supports algebraic operations more akin to those found in the integers.
• How does the structure of an integral domain allow for unique factorization, and why is this important?
  • The structure of an integral domain allows for unique factorization because it does not contain zero divisors, which helps ensure that every non-zero element can be represented as a product of irreducible elements in a consistent manner. This is crucial in number theory and algebra since it enables mathematicians to break down complex equations into simpler parts. Unique factorization is fundamental for solving equations and analyzing polynomial behavior within the domain.
• Evaluate the implications of having a characteristic of an integral domain be either zero or prime. How does this affect its overall structure?
  • The characteristic of an integral domain being zero implies that there are no finite cycles in addition, resembling the integers where you can keep adding without looping back. If the characteristic is prime, it indicates that adding the unity element repeatedly will eventually reach zero after a finite number of steps. This impacts how elements interact within the domain and influences the types of substructures present, including whether certain linear combinations can generate other elements or if specific homomorphisms can be established with other rings or fields.

Related terms

Ring:

A set equipped with two operations, addition and multiplication, satisfying certain properties such as associativity, distributivity, and the existence of an additive identity.

Field:

A set in which addition, subtraction, multiplication, and division (except by zero) are defined and behave as they do in the rational numbers, allowing for the existence of multiplicative inverses.

Zero Divisor: An element 'a' in a ring such that there exists a non-zero element 'b' in the ring where the product 'a * b = 0'.