Hiram Paley's A First Course in Abstract Algebra PDF

By Hiram Paley

ISBN-10: 0030549655

ISBN-13: 9780030549656

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12 suggest that the endomorphism ring of a finite rank torsion free group can be complicated. known as C'orner's Theorem, dimensional for each i. there is a rank 3 torsion free has only finitely many non-equivalent Examples A m = n and there is a A i = Bo(i) It is proved in Theorem ii. Ii that if free group then if of In fact, the following theorem, shows that essentially every subring of a finite Q-algebra may be realized as the endomorphism ring of a finite rank torsion free group. 13: Suppose that R is a ring and that the additive group of R is reduced torsion free of rank free group A of rank Proof: Fuchs Corner 2n [ 7 ], Vol.

0 1,j qij ~ 0 so that b! i,3. ~ Zp for is a unit 1 ~ j ~ m, is a unit and o f Q. since Perform B = Zpb~ 9 ... 9 Zpb~ 9 for some I c n +, noting that > i. ///. -Let k 2 with each = 0 F = A/D). (Qam+ 1 e ... 9 Qan), and c Q. Then is a free B = B 1 $ B 2 = (Zpb I $ ... m Zpbk) $ (Qbk+ 1 m ... 9 Qb~), qi,j space. 3 is necessary. be the class of pure subgroups H of finite rank completely the class of torsion free homomorphic images of 43 finite rank completely decomposable torsion free class containing Then groups, and let R be the smallest the class of all rank-i torsion free groups.

N}, pure subgroup of A with rank(B) Proof. is torsion free of finite rank > 1 a 2 is a pure rank-i subgroup of let A'. If A(I) = Z {Aili ~ I}. then A' = A 1 9 ... $ A n ~ A + 0 is a pure subgroup of B = E {BnA(1)[I Thus, be an epimorphism. it is sufficient A. If B For is a ~ n+}. Then B' = Z (B' n A'(I) II ~ n +} B = f(B') = E (B n A(1)II ~ n+}. and that B' = f-l(B) then to assume that A = A 1 9 ... 9 A n . If p is a prime then B is a pure subgroup of A 9 P P sufficient to assume that A is p-local since if B = Therefore, it is S {Bp n Ap(I) II ~ n +) P then B = ~ Bp = Since E {B n A(I)II ~ n+}.

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A First Course in Abstract Algebra by Hiram Paley


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