Definition: Principal Ideal
Let \(I\lhd R\) be an left (or right) ideal of a integral domain $(R, + , \cdot).$
 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.
 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.
 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
 Proposition: Principal Ideal Generated by A Unit
 Proposition: Criterions for Equality of Principal Ideals
 Lemma: Divisibility of Principal Ideals
 Definition: Principal Ideal Ring
 Definition: Principal Ideal Domain
Mentioned in:
Definitions: 1 2
Proofs: 3 4 5 6 7
Propositions: 8 9 10 11
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References
Bibliography
 Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück