172 III More on Divisibility
impliesxk =ykdkfor someyk ∈ Rif 1≤k≤randxk =0ifr <k ≤n.In
particular,xka′k=Ofor 1≤k≤n, and thus the moduleMis the direct sum of the
submodulesM′ 1 ,...,M′ngenerated bya′ 1 ,...,a′nrespectively.
IfNkdenotes the set of allx∈Rsuch thatxa′k=O,thenNkis the principal ideal
ofRgenerated bydkfor 1≤k≤randNk ={ 0 }forr<k≤n. The divisibility
conditions on thed′s imply thatNk+ 1 ⊆Nk( 1 ≤k<r).IfNk=Rfor somek,then
a′kcontributes nothing as a generator and may be omitted.
Evidently the submoduleM′generated by a′ 1 ,...,a′r consists of alla ∈ M
such thatxa=Ofor some nonzerox ∈ R, and the submoduleM′′generated by
a′r+ 1 ,...,a′nhasa′r+ 1 ,...,a′nas a basis. Thus we have proved thestructure theorem
for finitely generated modules over a principal ideal domain:
Proposition 42Let R be a principal ideal domain andMa finitely generated
R-module. ThenMis the direct sum of two submodulesM′andM′′,whereM′consists
of alla∈Msuch that xa=Ofor some nonzero x∈R andM′′has a finite basis.
Moreover,M′is the direct sum of s submodules Ra 1 ,...,Ras, such that
0 ⊂Ns⊆···⊆N 1 ⊂R,where Nkis the ideal consisting of all x∈R such that xak=O( 1 ≤k≤s).
The uniquely determined submoduleM′is called thetorsion submoduleofM.The
free submoduleM′′is not uniquely determined, although the number of elements in a
basis is uniquely determined. Of course, for a particularMone may haveM′={O}or
M′′={O}.
Any abelian groupA, with the group operation denoted by+, may be regarded as a
Z-module by definingnato be the suma+···+awithnsummands ifn∈N,to
beOifn=0, and to be−(a+···+a)with−nsummands if−n∈N.Thestruc-
ture theorem in this case becomes thestructure theorem for finitely generated abelian
groups: any finitely generated abelian groupAis the direct product of finitely many
finite or infinite cyclic subgroups. The finite cyclic subgroups have ordersd 1 ,...,ds,
whered 1 >1ifs>0anddi|djifi≤j. In particular,Ais the direct product of a
finite subgroupA′(of orderd 1 ···dr), itstorsion subgroup,andafreesubgroupA′′.
The fundamental structure theorem also has an important application to linear
algebra. LetVbe a vector space over a fieldKandT :V →Va linear transfor-
mation. We can giveVthe structure of aK[t]-module by defining, for anyv∈Vand
anyf=a 0 +a 1 t+···+antn∈K[t],
fv=a 0 v+a 1 Tv+···+anTnv.IfVis finite-dimensional, then for anyv∈Vthere is a nonzero polynomialfsuch
thatfv=O. In this case the fundamental structure theorem says thatVis the direct
sum of finitely many subspacesV 1 ,...,Vswhich are invariant underT.IfVihas
dimensionni≥1, then there exists a vectorwi∈Visuch thatwi,Twi,...,Tni−^1 wi
are a vector space basis forVi( 1 ≤i ≤s). There is a uniquely determined monic
polynomialmiof degreenisuch thatmi(T)wi=Oand, finally,mi|mjifi≤j.
The Smith normal form can be used to solve systems of linear ordinary differential
equations with constant coefficients. Such a system has the form