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Modules Over a Principal Ideal Domain 157

(6.4 )
“ c “ Ä “ “

or equivalently,
ann²+ ³ ‹ ann²+ ³ ‹ Ä

The numbers are called invariant factors of the decomposition.

For instance, in the example above suppose that the elementary divisors are

Á Á Á Á Á

Then the invariant factors are

~

~

~
The process described above that passes from a sequence Á of elementary

(
)
(
)
divisors in order 6.3 to a sequence of invariant factors in order 6.4 is
(
satisfying 6.4 , )
reversible. The inverse process takes a sequence Á Ã Á
factors each into a product of distinct nonassociate prime powers with the

primes in the same order and then “peels off” like prime powers from the left.
(The reader may wish to try it on the example above. )
This fact, together with Theorem 6.4, implies that primary cyclic
decompositions and invariant factor decompositions are essentially equivalent.
Therefore, since the multiset of elementary divisors of 4 is unique up to
associate, the multiset of invariant factors of 4 is also unique up to associate.
Furthermore, the multiset of invariant factors is a complete invariant for
isomorphism.
(
Theorem 6.16 The invariant factor decomposition theorem) Let 4 be a
finitely generated torsion module over a principal ideal domain . Then
9
4~ + l Ä l +
where D is a cyclic submodule of 4 , with order , where

“ c “ Ä “ “
This decomposition is called an invariant factor decomposition of 4 and the
scalars are called the invariant factors of 4 .

1 The multiset of invariant factors is uniquely determined up to associate by
)
the module 4 .
)
2 The multiset of invariant factors is a complete invariant for isomorphism.
The annihilators of an invariant factor decomposition are called the invariant
ideals of 4 . The chain of invariant ideals is unique, as is the chain of