additive inverse of one element times another element is the additive inverse of their product
Let R be a ring. For all x,y∈R
(-x)⋅y=x⋅(-y)=-(x⋅y)
All we need to prove is that (-x)⋅y+x⋅y=x⋅(-y)+x⋅y=0
Now: (-x)⋅y+x⋅y=((-x)+x)⋅y by distributivity.
Since (-x)+x=0 by definition and for all y, 0⋅y=0 we get:
(-x)⋅y+x⋅y=0⋅y=0 and thus (-x)⋅y=-(x⋅y)
For x⋅(-y), use the previous properties of rings to show that
x⋅(-y)+x⋅y=x⋅((-y)+y)=x⋅0=0
and thus x⋅(-y)=-(x⋅y)
Title | additive inverse of one element times another element is the additive inverse of their product |
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Canonical name | AdditiveInverseOfOneElementTimesAnotherElementIsTheAdditiveInverseOfTheirProduct |
Date of creation | 2013-03-22 15:43:40 |
Last modified on | 2013-03-22 15:43:40 |
Owner | cvalente (11260) |
Last modified by | cvalente (11260) |
Numerical id | 8 |
Author | cvalente (11260) |
Entry type | Theorem |
Classification | msc 16-00 |
Classification | msc 20-00 |
Classification | msc 13-00 |