can this be proved without the use of Zorn's lemma? perhaps i am not understanding the question, but condition two looks like an ascending chain condition on the principal ideals:

.

i was thinking about this, and i ran into the following difficulty:

given any (non-zero) ideal I, we can form a chain of ideals by taking:

where

.

where

is chosen by finding

and setting

, so that

,

where

is chosen by finding

and setting

, so that

,

and so on, which by (2) terminates with some

.

now, it would be nice if we could conclude that

, but perhaps for some other choice of the

we would wind up with "a different N". that is, how do we know that we can find an N that makes every element in

an associate of

? (i think this is where Zorn's lemma comes in. if i understand what you wrote, each Ra in your answer, is one of my "

"s).

my apologies if i have stated this badly.