Abelian Groups - László Fuchs [Springer]

1 Freeness and Projectivity 79
Let fa i g i2I be a set of generators of a group A, and W F ! A an epimorphism
from a free group F D˚i2I hx i i such that x i D a i for each i 2 I. Ker consists of
those linear combinations m 1 x i1 CCm k x ik 2 F with integral coefficients m i for
which m 1 a i1 CCm k a ik D 0 holds in A. These equalities are called the defining
relations relative to the generating system fa i g i2I .
It follows that the group A is completely determined by giving a set fa i g i2I of
generators along with the set of all defining relations:
A Dha i .i 2 I/j m j1 a i1 CCm jk a ik D 0.j 2 J/i (3.2)
(since we are dealing exclusively with abelian groups, the commutativity relations
are not listed). Indeed, if (3.2) isgiven,thenA is defined as the factor group
F=H, whereF is a free group on the free set fx i g i2I of generators, and H is the
subgroup of F, generated by the elements m j1 x i1 C C m jk x ik for all j 2 J.
The relations between the given generators of A are exactly those which are listed
in (3.2), and their consequences. (The emphasis is on the non-existence of more
relations.) Equation (3.2)issaidtobeapresentation of A.
Example 1.10. A presentation of a free group F with free generators fx i g i2I is given as
F Dhx i .i 2 I/ j ¿i (there are no relations between the generators). Of course, there are numerous
other presentations; e.g. Z Dhx; y j 2x 3y D 0i.
Example 1.11. The group C Dhx j nx D 0i for n 2 N is cyclic of order n.
F Notes. The material on free groups is fundamental, and will be used in the future without
explicit reference. Though in homological algebra, projectivity is predominant, in abelian group
theory freeness seems to prevail. Fortunately, for abelian groups, freeness and projectivity are
equivalent, while for modules, the projectives are exactly the direct summands of free modules.
Projective modules are rarely free; they are free over principal ideal domains (but not even over
Dedekind domains that are not PID), and over local rings (Kaplansky [2]).
Theorem 1.6 holds for modules over left principal ideal domains. Submodules of projectives
are again projective if and only if the ring is left hereditary, i.e., all left ideals are projective.
Theorem 1.2 holds over commutative rings or under the hypothesis that at least one of and
is infinite. There exist, however, rings R such that all free R-modules ¤ 0 with finite sets of
generators are isomorphic. It is perhaps worthwhile pointing out that every R-module is free if and
only if R is a field, and every R-module is projective exactly if R is a semi-simple artinian ring.
The property that all R-modules have projective covers characterizes the perfect rings, introduced
by H. Bass.
Hausen [6] defines a group P -projective for an infinite cardinal if it has the projective
property with respect to all exact sequences 0 ! A ! B ! C ! 0 with jCj