# 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 $a\in R.$

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:

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