TITLE: Is the ideal generated by irreducible element in principal ideal domain maximal?
QUESTION [13 upvotes]: Possible Duplicate:
Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible 

I am trying to see whether the ideal generated by irreducible element in a principal ideal domain (PID) is maximal ideal.
Suppose r is irreducible in a PID say D
Let I be an ideal of D containing (r) the ideal generated by r 
Since  D is a principal ideal domain, there exist s in D such that I=(s), therefore (r) is a subset of (s).
So,  r=st , for some t in  D   but r is irreducible, this implies that s or t is a unit.
If s is a unit then I= (s)= D .
If t is a unit then (r)= I=(s). But I am not sure this is true, because I do not have reason for saying (r)= I=(s), so that I can conclude and say (r) is maximal.
I need a little help for this. Thanks

REPLY [4 votes]: An element is irreducible iff the ideal it generates is maximal amongst the principal ideals.
If all ideals are principal, then an element is irreducible iff the ideal it generates is maximal.