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