By Wella Brown, Julian Kitt

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**Sample text**

1 Homomorphisms of Abelian Groups A map of abelian groups ' W A ! 27) holds in the group B. 14 Verify that the composition of homomorphisms is again a homomorphism. 4 For every homomorphism of abelian groups ' W A ! 0/ D 0 and '. a/ D hold. a/ ( for all a 2 A/ B is a subgroup. 0/. a/C'. a C . 0/ D 0. 2 Kernel of a Homomorphism For every homomorphism of abelian groups ' W A ! 1 For every homomorphism of abelian groups ' W A ! b/. In particular, every nonempty fiber is in bijection with ker ', and ' is injective if and only if ker ' D 0.

9 An integer is prime if it is not equal to the product of two noninvertible integers. See Sect. 2 on p. 7. 10/, the product of nonzero classes Œ2 Œ5 D Œ10 equals zero. 8/, the nonzero element Œ2 has a zero cube: Œ23 D Œ8 D Œ0. In an arbitrary commutative ring K, a nonzero element a 2 K is called a zero divisor if ab D 0 for some nonzero b 2 K. Note that an invertible element a 2 K cannot divide zero, because multiplication of both sides of the equality ab D 0 by a 1 forces b D 0. In particular, a commutative ring with zero divisors cannot be a field.

We write U 6 W if U ' Œw/ for some w 2 W, and write U < W if U 6 W and U § W. As good training in the use of the principle of transfinite induction, I strongly recommend the following exercise. 17 For any two-well ordered sets U; W, either U < W or U ' W or W < U holds. Classes of isomorphic well-ordered sets are called cardinals. Thus, the set N can be identified with the set of all finite cardinals. All the other cardinals, including N itself, are called transfinite. 3 Zorn’s Lemma Let P be a poset.