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(3): Counter example for Monoids by Richard)

Actually, what controls being able to make inverses is not just the cancellation property.  What is known as the Ore condition is also needed, according to Wikipedia article http://en.wikipedia.org/wiki/Ore_condition

The example sought is thus a cancellation ring that does not satisfy the Ore condition.  Such would seem to exist, or otherwise the Ore condition would be baloney!

Posted by Richard on 2006-08-10 19:53:03