The problem involves showing that every ideal of the ring R, specifically when R is the integers modulo n (Zn), can be expressed in the form mR for some integer m. The discussion revolves around properties of ideals and the application of the division algorithm.
The discussion is active, with participants offering hints and suggestions regarding the use of the division algorithm and the structure of ideals. Some participants are exploring different interpretations of the problem, and there is a collaborative effort to clarify concepts without reaching a definitive conclusion.
There is a mention of the need to express ideals in a specific form and the constraints of working within the framework of Zn. Participants are also navigating the implications of the smallest element in an ideal and how it relates to the overall structure of the ideal.
VeeEight said:Use the division algorithm.
kimberu said:You mean, say that n = qd + r, or for m?
Sorry, I'm totally lost on this problem.
TMM said:Suppose you have an ideal of the form mR+nR. How can you express this as a principal ideal?
So from here...can I say that s must equal 0 and x = aq for all x, since otherwise it's a contradiction because a is the smallest element?VeeEight said:Let a be the smallest positive element in your ideal I. If you have some element x in I, then x = aq + s for some s less then a.
VeeEight said:Yes.