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 Another possibility? by e.g.)

You are not even using the fact that we are talking about a ring, ie. a structure in which we can add stuff. If your proof was correct (which I am afraid it isnt), you would have proven that right-cancellation is equivalent to left-cancellation in every group. This would be a much stronger statement than the one Richard gave.

I wonder if someone can give a counter example of Richards statement for groups...

Posted by JLo on 2006-07-21 07:46:33