Branches History Index

◀ ▲ ▶Branches / Algebra / Definition: Principal Ideal

Definition: Principal Ideal

Let \(I\lhd R\) be an left (or right) ideal of a integral domain $(R, + , \cdot).$

  1. If there is an \(a\in I\) with \(I=Ra:=\{ra\in I:\quad \forall r\in R\}\), we call with an left principal ideal.
  2. If there is an \(a\in I\) with \(I=aR:=\{ar\in I:\quad \forall r\in R\}\), we call with an right principal ideal.
  3. If \(I\) is both, a left and a right principal ideal, we call \(I\) simply a principal ideal (generated by $a$) and denote it by $(a).$

Loosely speaking, a principal ideal $(a)$ is a subset of the integral domain consisting of all multiples of a given element $a\in R.$

Table of Contents

  1. Proposition: Principal Ideal Generated by A Unit
  2. Proposition: Criterions for Equality of Principal Ideals
  3. Lemma: Divisibility of Principal Ideals
  4. Definition: Principal Ideal Ring
  5. Definition: Principal Ideal Domain

Mentioned in:

Definitions: 1 2
Proofs: 3 4 5 6 7
Propositions: 8 9 10 11

Thank you to the contributors under CC BY-SA 4.0!

Github:
bookofproofs
non-Github:
@Brenner

References

Bibliography

  1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013

Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück