Consider 〈S, +, •〉, where S is a set and + and • are binary operations on S such that
〈S, +〉 is a group,
〈S*, •〉 is a group where S* consists of all elements of S except the additive identity element, a(b + c) = (ab) + (ac) and (a + b)c = (ac) + (bc) for all a, b, c ∊ S.
Show that 〈S, +, •〉 is a division ring. [Hint: Apply the distributive laws to (1 + 1)(a + b) to prove the commu-tativity of addition.]
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