By Laszlo Fuchs
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Additional resources for Infinite Abelian Groups, Volume 1 (Pure & Applied Mathematics)
4. MODULES Most of the theorems in abelian group theory can be generalized mutatis rriutandis to unital modules over a principal ideal domain R with identity, and everything can be carried over-without any modification in the proofs -if R has the additional property that all quotient rings R/(a) with 0 # a E R are finite. , to describe the class of rings such that, for the modules over these rings, the theorem in question holds. , modules over the ring Z of integers. Occasionally, however, we have to consider modules, since they yield a natural method of discussion.
N, 9. DIRECT SUMMANDS We have called a subgroup B of A a direct summand of A if A = B @C for some C 5 A. For the projections IC : A -+ B, 0 : A C conditions (1) in 8 hold. Next we focus our attention on B. ] We then have the following useful lemma. 1. If there is a projection IC of A onto its subgroup B, then B is a direct summand of A . The map B = 1, - 71 is an endomorphism of A , satisfying conditions (1) in 8. Hence we have A = B @ BA. 0 A rather trivial criterion for B ( 5A) to be a direct summand of A is that the cosets of A mod B have representatives which form a subgroup C ( S A ) .
Ya. Kulikov, and T. Szele are significant. In the NOTES 35 late 1950’s the homological aspects began to play a stimulating role in abelian group theory. Theorem ( 1 . 1 ) is most elementary, but fundamental. In view of it, the structure theory of abelian groups splits into the theories of torsion and torsion-free groups, and investigations of how these are glued together to form mixed groups. 1). It should be noted that it does not generalize to arbitrary modules. If a “torsion,” element a of an R-module M is defined by o ( a ) # 0, then it is not true in general that the torsion elements form a submodule.
Infinite Abelian Groups, Volume 1 (Pure & Applied Mathematics) by Laszlo Fuchs