Page 167 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes
Example: Let X be a set and Y be a non-empty subset of X. Show that
I = { A   (x) | AY =  } is an ideal of  (X).
In particular, if we take Y = {x }; where X is a fixed element of X, then
0
0
I = { A   (X) | x  A } is an ideal of  (X).
0
Solution: Firstly,   1,
Secondly,  A, B E I,
(A – B )  Y = (A  B )  Y = (A  Y)  (B  Y ) =    = , so that A – B  I.
Finally, for A  I and B E (X),

(AB)  Y = (A  B)  Y = (A  Y)  B =   B = , So that AB  I
Thus, I is an ideal of  (X).

Example: Consider the ring C[0, 1]
Let M = ( f  C[0, 1] | f(1/2) = 0 ). Show that M is an ideal of C[0, 1].
Solution: The zero element 0 is defined by 0(x) = 0 for all x  [0, 1]. Since 0(1/2) = 0, O E M.

Also, if f, g  M, , then (f – g) (l /2 ) = f (1/2 ) – g (1/2 ) = 0 – 0 = 0.
So, f – g  M .
Next, iff  M and g  C [0, 1] then (fg) (1/2) = f(1/2) g (1/2) = 0 g(1/2) = 0, so f  M.
g
Thus, M is an ideal of C[0, 1].
When you study Unit 17, you will see that M is the kernel of the homomorphism

 : C[0, 1]  R : (f) = f(1/2).

Example: For any ring R and a , a  R, show that Ra + Ra = { x a + x a | x , x  R )
2
2
l
1
1 1
2
1
2 2
is an ideal of R.
Solution: Firstly, 0 = 0a t 0a .  0  Ra + Ra .
1
2
1
2
Next, (x a + x a ) – (y a + y a )
1 1
2 2
2 2
1 1
= ( x – y )a + (x – y )a  Ra + Ra  x , x , y , y  R.
1
1
2
2
2
1
1
1
2
2
2
1
Finally, for r  R and x a + x a  Ra + Ra ,
2 2
1
2
1 1
r(x a  x a ) = rx a + rx a  Ra + Ra . 2
1
2 2
1 1
2 2
1 1
Thus, Ra + Ra is an ideal of R.
1
2
This method of obtaining ideals can be extended to give ideals of the form { x a + x a + ... + x a
1 1
2 2
n n
| x  R } for fixed elements a ..,... , a, of R. Such ideals crop up again and again in ring theory. We
i
1
give them a special name.
Definition: Let a , ....., a, be given elements of a ring R. Then the ideal generated by a , ....., a,, is
1
1
Ra + Ra + ... + Ra = (x a + x a + ... + x a | x, E R ). a , ....., a,, are called the generators of this ideal.
1 l
n
n n
2 2
1
1
2
We also denote this ideal by < a , a , ....., an >
2
1
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