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?