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 re(2): Another possibility? LCPRCP in groups??? by Steve Herman)

Good point, Steve. Since groups have an inverse element, cancellation is obviously possible, either from left or right. I forgot that a ring is not necessarily a group with respect to multiplication. It is actually a monoid, i.e. a structure where

1. a(bc)=(ab)c for all a, b, c            (Associativity)
2. There is e such that ae=ea for all a   (Existence of unit element)

Can you come up with a counter example of Richards statement for monoids?

Posted by JLo on 2006-07-21 14:50:38