Abstract algebraLesson 1 of 2 in the course Rings and ideals

Integral Domains: Zero Divisors, Divisibility, and Fraction Fields

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

An integral domain is a commutative ring in which nonzero elements cannot multiply to zero. This lesson introduces the definition, explains why cancellation works, and builds familiarity with standard examples.

Throughout this unit, a ring is understood to have a multiplicative identity 11, and an integral domain is understood to be commutative.

A ring RR is an integral domain if it is nonzero and the product of any two nonzero elements is nonzero. Equivalently, there are no nonzero elements a,b∈Ra,b\in R such that ab=0ab=0. Such elements, when they exist, are called zero divisors.

All flashcards

What is a zero divisor in a commutative ring?
A nonzero element whose product with another nonzero element is zero.
What is a regular element of a ring?
An element for which multiplication by that element is an injective map.
What is a unit in a ring?
An element with a multiplicative inverse in the ring.
When are two elements associates in an integral domain?
When one is the product of the other and a unit.
What is the field of fractions of an integral domain?
The smallest field containing an isomorphic copy of the domain.

Quiz questions

  1. For a commutative ring RR with identity and an ideal PP, when is R/PR/P an integral domain?Answer: Exactly when PP is a prime ideal.
  2. If RR and SS are nonzero rings with identity, what is true of R×SR\times S?Answer: It is not an integral domain because (1,0)(0,1)=(0,0)(1,0)(0,1)=(0,0).
  3. Which statement about prime and irreducible elements in an integral domain is correct?Answer: Every prime element is irreducible, but an irreducible need not be prime.
  4. What can be concluded about a finite integral domain?Answer: It is a finite field.
  5. Which characteristic is possible for an integral domain?Answer: Zero or a prime number.

Based on the Wikipedia article Integral domain, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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