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 Another possibility? by e.g.)

e.g.: Your proof doesn't, I think, work.

You can't assume right-cancellation and then use that assumption to prove right-cancellation.  It is not good logic.

Logically, there are two ways to prove this:
(a) J Lo's approach: Prove that if x<> y, then xb = yb only if b = 0

(b) Assume right cancellation does not work, and prove that this leads to a contradiction.

Posted by Steve Herman on 2006-07-20 23:09:36