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): Counter example for Monoids by JLo)

I'm not really sure right now, but I think that the reason you only see the field of fractions being constructed in books for (commutative) integral domains is that the construction doesn't work in general in the noncommutative case. I need to do more research and cogitate more on this.  If you have any further thoughts, don't be shy. The basic suspicion is that the cancellation property is really just the existence of inverses lurking in the background. But it might possibly be that cancellation is a unique property that only becomes related to inverses through commutativity. Somebody has probably worked on this, but if they had no success one way or the other, we would not have heard about it probably. I no longer have access to any fancy professors of mathematics, so I can't just ask an expert. Books are disappointingly short on the subject of cancellation, but I guess that could  mean it isn't a very important thing to study.

Posted by Richard on 2006-08-07 18:07:12