Find two distinct factorizations of 10 in . Factor 10 as a product of four ideals in .
Evidently ; moreover, these factorizations are distinct since, as the only units in are , 2 is not associate to either of .
We claim that . To see the direction, note that this ideal product is generated by all possible selections of one generator from each factor. Name the generators such that . Now any selection which includes (without loss of generality) and does not include must include both and , and so this ideal product is in .
Note also that .