Page 37 - Advanced Linear Algebra
P. 37
Preliminaries 21
of all finite linear combinations of elements of , with coefficients in , is an
9
:
(
ideal in , called the ideal generated by . It is the smallest in the sense of set
:
9
)
:
9
inclusion ideal of containing . If : ~ ¸ ÁÃ Á ¹ is a finite set, we write
º ÁÃ Á » ~ ¸ b Ä b 9Á :¹
Note that in the previous definition, we require that have an identity. This is
9
to ensure that : º:» .
Theorem 0.20 Let be a ring.
9
1 The intersection of any collection ¸ ? 2 ¹ of ideals is an ideal.
)
) ? ? Ä is an ascending sequence of ideals, each one contained in
2 If
is also an ideal.
the next, then the union ?
)
3 More generally, if
9 ? ~¸ 0¹
is a chain of ideals in , then the union @ ~ 0 is also an ideal in .
9 ?
9
Proof. To prove 1 , let @ ) ? ~ . Then if Á @ , we have Á ? for all
2. Hence, c ? for all 2 and so c @ . Hence, is closed
@
under subtraction. Also, if 9 , then ? for all 2 and so @ . Of
)
)
course, part 2 is a special case of part 3 . To prove 3 , if Á @ ) , then ?
and ? for some Á 0 . Since one of and is contained in the other, we
?
?
may assume that ? ? . It follows that Á ? and so c ? @ and if
9, then ? @ . Thus is an ideal.
@
Note that in general, the union of ideals is not an ideal. However, as we have
just proved, the union of any chain of ideals is an ideal.
Quotient Rings and Maximal Ideals
:
Let be a subset of a commutative ring with identity. Let 9 be the binary
relation on defined by
9
¯ c :
It is easy to see that is an equivalence relation. When , we say that
and are congruent modulo . The term “mod” is used as a colloquialism for
:
modulo and is often written
mod :
As shorthand, we write .
