Suppose the vector space is the direct sum of cyclic -modules whose annihilators are , , , and . Determine the invariant factors and elementary divisors of .
The elementary divisors of are thus , , , , , , , and .
The invariant factors of are , , and .
If is reducible over , then is has a root , and indeed . Now by the Chinese Remainder Theorem for modules we have .
So the elementary divisors of are , , , , , , , , , and .
The invariant factors of are , , and . (Which are the same as if did not contain . This is to be expected in light of Corollary 18 on page 477 of D&F.)