 Apologies. I was remembering more than I was thinking: you are right in that the relation doesn't hold at the level of elements. The correct statement is`````` (2) = (1+i)^2 `````` where (2) and (1+i) represent the ideals generated by 2 and 1+i respectively. Officially, an ideal of a ring R is a subset I of R which is closed under addition and under and multiplication by elements of R:`````` i, j ∈ I implies i + j ∈ I i ∈ I implies ri ∈ I for all r ∈ R `````` For our purposes, a simpler case -- that of principal ideals -- suffices, and that's what I'll discuss in the rest of this comment.The ideal generated by an element a of a ring R, usually denoted (a) or aR, is essentially the subset of all multiples of a in R. For instance, over the integers, we have ideals like`````` (2) = {..., -4, -2, 0, 2, 4, ...} (3) = {..., -6, -3, 0, 3, 6, ...} (4) = {..., -8, -4, 0, 4, 8, ...} (6) = {..., -12, -6, 0, 6, 12, ...} `````` One can also define the ideal generated by the elements a,b,c,... of R:`````` (a,b,c,...) = {aa' + bb' + cc' + ... : a',b',c' ∈ R} `````` i.e. the "linear combinations" of a, b, c, and so on.Here are a few things you might like to verify (i.e. just nod along) about some ideals of Z, to get familiar with the notation:`````` (1) = Z (i.e. all the integers) (2) = (-2) (6) = (2) ∩ (3) 15Z = 3Z ∩ 5Z `````` where ∩ is the intersection of the two sets, or the set of common elements. ("Fact": The intersection of two ideals is also an ideal.)In general, if you have an ideal of the form (a), then we always have`````` (a) = (ua) for all units u `````` Write G (for Gauss?) for Z[i]. In G, the previous statement means that (2) = (2i). To check this, note that we have`````` x ∈ (2) iff x = 2y for some y iff x = (2i)(-yi) iff x ∈ (2i) `````` so all elements of (2) are in (2i), and vice versa.Now one defines the product of ideals: given ideals I and J in some ring R,`````` IJ = ideal generated by the elements {ij : i ∈ I, j ∈ J} `````` In our case, if I = (a) and J = (b), IJ is just (ab). Question: what's the relation between (ab) and (a) ∩ (b)?For instance, you can verify that`````` (2)(3) = (6) `````` and, to bring us back to our original point,`````` (1+i)(1-i) = (2i) = (2) = (1+i)^2 `````` where all the products are ideal products.--There is a very incomplete treatment at  (with a bit of the LaTeX broken) that is meant to provide background for a sequence of posts on algebraic geometry -- it's been suspended for a while, but I'm going to look over a couple of drafts this week!: As always in math: if you can't prove it, just add a few adjectives or simply change the language. Facts arrived at by alternative definitions, if you will. :) Search: