In my current algebraic number theory course we have defined the multiplication of 2 ideals as the smallest ideal containing all products of elements of both,
[i.e: let I and J be ideals of a ring R, then IJ:={$\sum_1^k a_ib_i : a_i\in I , b_i\in J$} where k depends on R ]
we haven't however, formally defined ideal powers, by which I mean ideals of the form $I^x$ where x is an integer.
However, using the first, informal definition, it seems to me that all products of elements from I will just again form I
i.e. using the formal definition, and taking x = 2 I find that
$I^2$ = II = {$\sum_i^k a_ib_i : a_i\in I , b_i\in I$} = {$\sum_1^k c_i : c_i\in I$} = {$d : d\in I$} = I
and clearly extending this definition to higher powers results in the same thing?
Thanks for any clarification!