Definition: A ring is a set R, equipped with two operations 1, Addition: + : R x R - R 2. Multiplication: . : R x R - R, such that the following axioms are satisfied: 1. For all a, bE R, a + b= btu. 2. For all a, b, cell at (bte) = (at b) +c. 3. There exists an clement 0 6 R such that for all a c R, 0 + a = a. We call 0 the additive identity of R. 4. For all a e R there exists an element. be R such that a + b= 0. We write b = -o and call -a the additive inverse of a. 5. For all o, bee R (able - a(be). 6. For all a, b, c ( R o(b + c) = ab t ac and (b + cja = be + ca. Definition Let R be a ring. We say R is unital if there exists le R such that lo = al = a for all a c R. We call 1 the multiplicative identity (or unity) of R Definition Let R be a ring. We say R is commutative If ob = bo for all a, be R. Definition Let # be a commutative and unital ring. We say 0 # a e R is a unit in R if there exists be R with oh - 1. Definition Let A be a commutative and unital ring with at least two elements. We say R is a field if and only if every nonzero element of R is a unit. Let R = Z[v2- (a) Prove that 1 4 v2 is a unit of R. (b) Prove that 1 4 v2 is the smallest unit of A which is greater than I. (Hint: If a+ dvd ) 1 and |(a + bva)(a - dvd)| = 1, then -1