Page 171 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 155

Elementary Divisors
Since the chain of annihilators

Á
ann²ºº# »»³ ~ º »
Á

Á
is unique except for order, the multiset ¸ ¹ of generators is uniquely

determined up to associate. The generators Á are called the elementary

divisors of 4 . Note that for each prime , the elementary divisor Á of largest

exponent is precisely the factor of ²4³ associated to .
Let us write ElemDiv²4³ to denote the multiset of all elementary divisors of
4. Thus, if  ElemDiv ²4³, then any associate of is also in ElemDiv ²4³.
We can now say that ElemDiv²4³ is a complete invariant for isomorphism.
Technically, the function 4ª ElemDiv ²4³ is the complete invariant, but this
hair is not worth splitting. Also, we could work with a system of distinct
representatives for the associate classes of the elementary divisors, but in
general, there is no way to single out a special representative.
Theorem 6.14 Let be a principal ideal domain. The multiset ElemDiv ² 4 ³ is
9
a complete invariant for isomorphism of finitely generated torsion -modules,
9
that is,
4 š 5 ¯ ElemDiv ²4³ ~ ElemDiv ²5³
We have seen (Theorem 6.2) that if

4~ ( l )
then
²4³ ~ lcm ² ²(³Á ²)³³
Let us now compare the elementary divisors of 4 to those of and .
(
)
Theorem 6.15 Let 4 be a finitely generated torsion module over a principal
ideal domain and suppose that
4~ ( l )
)
1 The primary cyclic decomposition of 4 is the direct sum of the primary
cyclic decompositons of and ; that is, if
(
)
( ~ ºº »» and ) ~ ºº »»
Á
Á
are the primary cyclic decompositions of and , respectively, then
(
)
M ~ 4 ºº »» l 4 Á 5 ºº »»5 Á
is the primary cyclic decomposition of M.