Question: If (A, m) is a discrete valuation ring, prove that m = (x) for some x ∈ A and that every non-zero ideal of A is a power of m ? Let A be a domain with quotient field K. Prove that the set given by G = {xA| x ∈ K∗} is an ordered abelian group with the usual multiplication of fractional ideals and where G is ordered by xA ? yA if and only if xA ⊃ Ya ? Let K = Z2 be the field with two elements and let F = ⊕i∈NFi, where Fi = K for all i. Prove that F∗ = HomK(F, K) is uncountable and that F is not a reflexive K-module.