Algebra I: Textbook for Students of Mathematics by Alexey L. Gorodentsev (auth.)

By Alexey L. Gorodentsev (auth.)

This ebook is the 1st quantity of a radical “Russian-style” two-year graduate direction in summary algebra, and introduces readers to the elemental algebraic constructions – fields, earrings, modules, algebras, teams, and different types – and explains the most rules of and techniques for operating with them.

The path covers vast components of complicated combinatorics, geometry, linear and multilinear algebra, illustration thought, classification thought, commutative algebra, Galois conception, and algebraic geometry – themes which are usually missed in general undergraduate courses.

This textbook is predicated on classes the writer has carried out on the self sustaining collage of Moscow and on the school of arithmetic within the greater university of Economics. the most content material is complemented via a wealth of workouts for sophistication dialogue, a few of which come with reviews and tricks, in addition to difficulties for self sustaining study.

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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: Œ23 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.

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