Page 43 - Advanced Linear Algebra

Preliminaries 27

Definition We will refer to the equivalence classes under the relation of being
associate as the associate classes of .
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Theorem 0.28 Let be a ring.
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1 An element "9 is a unit if and only if º"»~9 .
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2 — if and only if º » ~º » .
3 ) divides if and only if º » ‹ º . »

%
4 ) properly divides , that is, ~ % where is not a unit, if and only if
º » ‰ º ».
In the case of the integers, an integer is prime if and only if it is irreducible. In
any integral domain, prime elements are irreducible, but the converse need not
hold. In the ring { j c µ ~ ¸ b j c “ Á  {´ ¹ the irreducible element
(
divides the product ² b j c ³² c j c ³ ~ but does not divide either
factor.)

However, in principal ideal domains, the two concepts are equivalent.

Theorem 0.29 Let be a principal ideal domain.
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1 An  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.
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3 The elements Á  9 are relatively prime , that is, have no common
nonunit factors, if and only if there exist Á  9 for which
b ~
This is denoted by writing ² Á ³ ~ .
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Proof. To prove 1 , suppose that is irreducible and that º » ‹º » ‹9 . Then

 º » and so ~% for some %  9. The irreducibility of implies that or
% is a unit. If is a unit, then º »~9 and if % is a unit, then º »~º% »~º ».
(

This shows that º » is maximal. We have º » £ 9 , since is not a unit.)

Conversely, suppose that is not irreducible, that is, ~ where neither nor

is a unit. Then º » ‹º » ‹9. But if º »~º », then — , which implies that

º is a unit. Hence » £ º » º » ~ 9 . Also, if , then must be a unit. So we
conclude that º » is not maximal, as desired.
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To prove 2 , assume first that is prime and ~ . Then “ or “ . We

may assume that “ . Therefore, ~ % ~ % . Canceling 's gives ~ %
and so is a unit. Hence, is irreducible. Note that this argument applies in
(

any integral domain.)

Conversely, suppose that is irreducible and let “ . We wish to prove that
“ or “ . The ideal º » is maximal and so º Á » ~ º » or º Á » ~ 9. In the
former case, “ and we are done. In the latter case, we have
~% b &