Page 36 - Advanced Linear Algebra
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20 Advanced Linear Algebra
The condition that a subring have the same multiplicative identity as is
:
9
required. For example, the set of all d matrices of the form
:
(~ > ?
(
for - is a ring under addition and multiplication of matrices isomorphic to
- :). The multiplicative identity in is the matrix ( , which is not the identity 0
of C . Hence, is a ring under the same operations as C Á ²-³ Á ²-³ but it is
:
not a subring of C Á ²-³ .
Applying the definition is not generally the easiest way to show that a subset of
a ring is a subring. The following characterization is usually easier to apply.
Theorem 0.19 A nonempty subset of a ring is a subring if and only if
9
:
) of is in
1 The multiplicative identity 9 9 :
)
2 : is closed under subtraction, that is,
Á : ¬ c :
)
3 : is closed under multiplication, that is,
Á : ¬ :
Ideals
Rings have another important substructure besides subrings.
Definition Let be a ring. A nonempty subset of is called an ideal if
?
9
9
1 ? ) is a subgroup of the abelian group 9 , that is, is closed under
?
subtraction:
Á ? ¬ c ?
)
2 ? is closed under multiplication by any ring element, that is,
?
Á 9 ¬ ? and ?
?
Note that if an ideal contains the unit element , then ~ ? 9 .
Example 0.13 Let ²%³ be a polynomial in -´%µ . The set of all multiples of
²%³,
º ²%³» ~ ¸ ²%³ ²%³ ²%³ -´%µ¹
is an ideal in -´%µ , called the ideal generated by ²%³ .
Definition Let be a subset of a ring with identity. The set
:
9
º:» ~ ¸ bÄb 9Á :Á ¹
