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posted by  fall3nm0nk on 11/3/2009 4:38:55 PM  |  status: Closed  |  Earned Karma: 1075

Rings and ideals

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Question Details:
Let R be a commutative ring with ideals I and J
If 
show that

Let R be a non-commutative ring with 2 sided ideals I and J
If 
show that
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posted by tdv (MNV) on 11/5/2009 6:56:47 AM  |  status: Live
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Response Details:
1. Let R be a commutative ring.
 I+J  = R impleise ther exists x in I and y in J such that x + y = 1.

Let  z ε IJ Then we have  z = xz+yz.
  Considering z J we have that  xz IJ and 
    considering z  I, we have that yz IJ 
Therefore xz + yz IJ
This proves z IJ.
Hence IJ IJ.
Let xy IJ. since x I and I is an ideal xy I.
  Similarly since y J and J is an ideal xy  J.
Hence xy IJ. 
So IJ   IJ
So we proved IJ = IJ.

2. Let R be a commutative ring.
 I+J  = R implies there exists x in I and y in J such that x + y = 1.

Let  z ε IJ Then we have  z = xz+yz.
  Considering z J we have that  xz IJ and 
    considering z  I, we have that yz JI 
Therefore xz + yz IJ+JI
This proves z IJ+JI.
Hence IJ IJ+JI.
Let xy IJ. since x I and I is an ideal xy I.
  Similarly since y J and J is an ideal xy  J.
Hence xy IJ. 
So IJ   IJ.
  Similarly we can show JI  IJ.
That is IJ + JI  IJ.
So we proved IJ+JI = IJ.
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