By L. E. Sigler (auth.)

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There can be therefore no function g: Y --+ X such that gof = Ix. Next suppose f is not a surjection. There exists yE Y such that f(x) =I= y for any x E X. If there were a function h: Y --+ X such that f o h = I y we would have (f o h)(y) = y. But then h(y) is an element of X such that f(h(y)) = y. This contradicts the second sentence of this paragraph. There can be no function h: Y--+ X such that f oh = Iy. Suppose f is not a bijection. Then either f is not an injection or f is not a surjection.

A) Some integral domains are fields. (B) Some division rings are fields. (C) Some fields are division rings. (D) Some integral domains are not fields. (E) Some division rings are not integral domains. 12. The set of even integers with the usual sum and product (A) is an integral domain (B) has no nontrivial divisors of zero (C) is a field (D) is a commutative ring (E) is a division ring. 13. The product ring Z x Z (A) is a commutative ring (B) is a commutative unitary ring (C) is an integral domain (D) is a field (E) None of the alternatives completes a true sentence.

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