Question

Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all i}.

It turns out that R forms a ring under the operations:

(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···), (a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)

Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}. You may use without proof the fact that I forms an ideal of R.

a) Is I principal in R? Prove your claim.

b) Is I prime in R? Prove your claim.

c) Is I maximal in R? Prove your claim. (HINT: For maximality, do not try to prove/disprove maximality directly. Instead, think about properties of maximal ideals from class that must hold true and apply them to this ideal.)

Answer 1