Page 155 - Advanced Linear Algebra

Chapter 6

Modules over a Principal Ideal Domain

We remind the reader of a few of the basic properties of principal ideal
domains.

Theorem 6.1 Let be a principal ideal domain.
9
)
1 An element  9 is irreducible if and only if the ideal º » is maximal.
2 An element in is prime if and only if it is irreducible.
)
9
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3 9 is a unique factorization domain.
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4 9 satisfies the ascending chain condition on ideals. Hence, so does any
finitely generated -module 4 . Moreover, if 4 is -generated, then any

9
submodule of 4 is -generated.

Annihilators and Orders
When is a principal ideal domain, all annihilators are generated by a single
9
element. This permits the following definition.
Definition Let be a principal ideal domain and let 4 be an -module.
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9
1 If is a submodule of 4 , then any generator of ann 5 ² ³ is called an order
)
5
of .
5
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2 An order of an element #4 is an order of the submodule ºº#»» .
For readers acquainted with group theory, we mention that the order of a
module corresponds to the smallest exponent of a group, not to the order of the
group.
Theorem 6.2 Let be a principal ideal domain and let 4 be an -module.
9
9
)

1 If is an order of 5 4 , then the orders of 5 are precisely the
5

associates of . We denote any order of by ² 5 ³ and, as is customary,
5
refer to ²5³ as “the” order of .
)
2 If 4~ ( l ) , then
²4³ ~ lcm ² ²(³Á ²)³³