Let T be the set of all sequences {an} of elements of Z. Prove the follow- ing. (i) T is an integral domain with respect to addition and multiplication defined by for all {an}, {bn} ET, {an} + {bn} = {an} {bn} = {en}, {an + bn} . where cn = Cizo aibn-i. (ii) To = {{an} € T | ai = 0 for all but a finite number of indices} is a subring with identity. (iii) The element (1,1,0,...) is a unit in T, but not in To. (iv) (2, 3, 1, 0, 0,...) is irreducible in T, but not in To.

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