By L. E. Sigler (auth.)

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X such that fah = ly if and only if f is a surjection; and (c) there exists k: Y--. X such that k of = Ix and f o k = ly if and only if f is a bijection. 6 Composition of functions PROOF. Suppose f is not an injection. Then there exist x 1, x 2 EX such that x 1 =I= x 2 and f(x 1) = f(x 2). If there were a function g: Y--+ X such that go f = Ix then (go f)(x 1 ) = x 1 and (go f)(x 2) = x 2. This means g(f(x 1 )) =I= g(f(x 2) ). But f(x 1 ) = f(x 2). This contradicts condition 2 in the definition of a function for g.

Give an example of a ring ( R, +, ·, 8) such that xy = - yx for all x, y e R yet it is false that x + x = 8 for all x e R. 18. Give an example of a ring (R, +, ·, 8) such that x + x = 8 for all x e R yet it is false that xx = x for all x e R. 19. Let (R, +, ·. 8, v) be a unitary ring. Let x be an element of R with a unique left multiplicative inverse. Prove x has an inverse in R. 20. An important example of a division ring which is not a field is given by the set of quaternions, {a + bi + cj + dkia, b, c, de IR}.

X 7l. = {(x, y)ix E 7l. }. We must define two binary operations on the set 7l. , calling the first addition and the second multiplication. (s, t) + (u, v) (s, t) · (u, v) = = (s + u, t (su, tv). + v), The operation + is an associative and commutative operation with a neutral element (0, 0). We verify these assertions. {s, t) + [(u, v) + (u, v) = {0, 0) + {s, t) = (s, t) + {0, 0) = (s, t) With respect to (s, t) 42 + (w, x)] (s, t) + (u + w, v + x) = (s + (u + w), t + (v + x)) = ( (s + u) + w, (t + v) + x) = (s + u, t + v) + (w, x) = [(s, t) + (u, v)] + (w, x).