- #1
dndod1
- 10
- 0
Homework Statement
Using only the ring axioms, prove that in a general ring (R, +,X)
aX (x-z) = (aXx)- (aXz) where all a,x,z are elements of R
Homework Equations
Group axiom 3: G3= There is an inverse for each element g^-1 *g =e
Ring axiom 3: R3= Two distributive laws connect the additive and multiplicatie structures.
For any x,y,z xX(y+z) = (xXy)+ (xXz)
and (x+y) X z= (xXz) + (yXz)
The Attempt at a Solution
My attempt. I thought that this would actually be straight forward; that I would just need to put -z as the addition of its inverse. I expected the rest to just fall into place.
Here's what I did:
aX (x-z) = (aXx)- (aXz)
Left hand side aX (x-z)
= aX(x + z^-1) from G3
= (aXx) + (aXz^-1) from R3
= (aXx)+ (aX -z) from G3
= (aXx)- (aXz), as required
I'm not sure whether I am allowed to just write the last line or whether I have left out some all important step!
Thank you very much in anticipation of your assistance.
Last edited: