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)

You and Steve Herman have posted some interesting stuff on monoids.  Thanks to both of you for paying some attention to the general subject of this problem. Getting back specifically to rings again, however, I have been wondering if there are any non-commutative cancellation rings that are not subrings of a division ring. Here, I would not necessarily require the ring to have a unity element, but one that does would be more interesting perhaps. That is, I usually include what some call rngs among what I call rings.

Posted by Richard on 2006-08-06 17:32:47