1. The problem statement, all variables and given/known data 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 2. Relevant 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) 3. 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.