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By V. A. Krechmar

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Dieses Buch ist eine moderne Einführung in die Algebra, kompakt geschrieben und mit einem systematischen Aufbau. Der textual content kann für eine ein- bis zweisemestrige Vorlesung benutzt werden und deckt alle Themen ab, die für eine breite Algebra Ausbildung notwendig sind (Gruppentheorie, Ringtheorie, Körpertheorie) mit den klassischen Fragen (Quadratur des Kreises, Auflösung durch Radikale, Konstruktionen mit Zirkel und Lineal) bis zur Darstellungstheorie von endlichen Gruppen und einer Einführung in Algebren und Moduln.

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Quigley [62] studied field extensions K D F. If a € K\F, then by Zorn's Lemma, there exists a subfield M 2 F that is maximal without a, that is, M(a) is the intersection of all subfields L of K properly containing M. 33 QUIGLEY'S extension of M. LEMMA 1. If M is maximal without a, then K is an algebraic We observe first that a is algebraic over M; for if a 2 ^ M, then M(a ) D M(a) D M(a2), so that a2 is algebraic over M. Let M be the algebraic closure of M in K. If 3n £ M, then 77 is transcendental over M, as is every element of M(rj) not in M.

Z) is countable. Cf. B. 19. Specker-Nobeling-Balcerzyk Theorems. For any cardinal a, the Specker group of Z a is the set of all bounded sequences. 19A T H E O R E M ( S P E C K E R is free for any cardinal a. [50]-N6BELING [68]). 4. 19B T H E O R E M Hom(Za,Z)is free. (SPECKER [50]-BALCERZYK [62]). g. , any subgroup A with \A\ < tti is free. 20. Dubois' Theorem. Let P(R) = Ru denote the set of all countable sequences of elements of R and B(R) those sequences such that the entries form a finite set.

23 T H E O R E M (CAMILLO [85]). If a direct sum M = ®aeAMa of right Rmodules splits off in their direct product YlaeA^a, then there is a cofinite subset B of A, such that R/ annR(®heBMh) satisfies the ace on annihilator right ideals. Cf. 7. Sarath and Varadarajan [74] characterized direct sum splitting very differently. 24A T H E O R E M (LENZING [76]). If the direct sum of infinitely many copies of a faithful R-module M splits off in their direct product then R satisfies the ace on annihilator right ideals (= ace _L).

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A Problem Book in Algebra by V. A. Krechmar


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