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Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai...

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.

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