Yes i think it is true in all 3 cases because we know I is an ideal and by the definition of ideal i think that all these cases must hold.
Thanks for your help in showing me this.
A further question i would appreciate your help with is this:
suppose b
R is an associate of a.
How do i prove that (I:b) = (I:a) ?
I started off by saying that a,b
R are associates if b=ua for some unit u
R.
I tried substituting this in to (I:b)
So (I:b) = {r
R : br
R}
= {r
R : uar
R} for some unit u
R
but i cant see how to proceed for here to try to make it equal (I:a).
Do you have any suggestions?