A ring is an algebraic system that supports unlimited addition, subtraction, and multiplication, with all the familiar laws (such as the distributive laws a(x+y)=ax+ay and (x+y)b=xb+yb) holding except that there may possibly be a,b pairs for which ab=ba does not hold. The ordinary integers are an example of a ring (where, however, ab=ba does always hold).

A ring has the left-cancellation property if ax=ay implies x=y for all nonzero a and all x and y, and has the right-cancellation property if xb=yb implies x=y for all nonzero b and all x and y.

Your mission should you choose to accept it: Prove that a ring has the left-cancellation property if and only if it has the right-cancellation property.

(In reply to Counter example for Monoids by JLo)

Sorry, JLo, but I believe that your counter-example is not associative, and therefore not a monoid, and therefore not a counter-example.

((a,b)*(0,1))*(1,c) = (a,b+1)*(1,c) = (a+1,b+1)

but

(a,b)*((0,1)*(1,c)) =  (a,b)*(1,1) = (a+1,b)

Posted by Steve Herman on 2006-08-06 23:55:16