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: Counter example for Monoids by Richard)

Don't know.

I am sure you know that for commutative rings R the example you are seeking for cannot exist, since you can extend every commutative cancellation ring to the field (=commutative division ring) of formal fractions p/q with p and q in R. Formally, you define an equivalence relation p/q ~ a/b :<=> pb=aq, then you "divide" the ring by this relation and show everything is well-defined...

So I suppose you have tried this process on non-commutative rings and it does not work?

Posted by JLo on 2006-08-07 16:43:52