What is the Proof that Every Ideal of Zn is Principal?

[WannaBe22]

Homework Statement

1.Prove that every ideal of Zn is principal

Homework Equations

The Attempt at a Solution

In 1-I've proved that if K is an ideal of Zn that contains an element k then it contains all the elements of the form mk (m in Zn)...But how can I prove that there are no elements that are not of the form mk?


Thanks in advance...

 

[rasmhop]

Let d be the least positive integer in your ideal K. You know all integers of the form md where m is an integer is in your ideal. To prove these are the only ones assume there is some integer d' in K that is not of the form md. Then we use the division algorithm to write it on the form:
d' = qd + r
for some integer q and 0<r<d. Now can you show that r is in K? If you can you will have a contradiction since r is positive and less than d.
(this general approach works for Euclidean domains in general, and shows that Euclidean domains are principal ideal domains)

 

[WannaBe22]

Well... if d' is in our ideal K then d'-qd is also in our ideal (since an ideal is also a sub-ring)... But d'-qd=qd+r-qd=r... Hence r must also be in our ideal and then we get a contradiction :)

THanks a lot!