This is a little self-portrait, "The Artist Trying to Learn Abstract Algebra", probably of no interest to anyone else.
I read an introduction to rings (in Gallian, fifth edition, which I enthusiastically recommend). Now I'm trying to come up with some conjectures and prove or disprove them before I start on the exercises. (This book has great exercises, and doesn't bother teaching anything in the text that it can teach in an exercise.)
I figured maybe every ideal of the ring Z[i] is a principal ideal generated by some element of the ring. This morning I think I have the proof. It's a consequence of Z[i] being enough like the integers to support Euclid's algorithm.
That in turn is a consequence of Z[i] having something like integer division. You can define a well-ordered metric M on Z[i] such that M(0) < M(a) where a is any other element; and for any a and nonzero b, there exist a quotient q and remainder r such that a = bq + r and M(r) < M(b). That the domain of M is well-ordered implies that Euclid's algorithm terminates.
Z[i] also has something like prime and composite elements. For example, 5+i can be factored into (1-i)(2+3i). I wonder if these two properties are actually the same thing.
I think the ideals of Z[i] generated by "prime" elements are prime ideals.