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.

Suppose we have Right Cancellation: xb=yb implies x=y. Multiplying by non-zero a, axb=ayb. By associative property, (ax)b=(ay)b. By Right Cancellation, we must have ax=ay. So, from xb=yb we get to ax=ay, requiring x=y.

Posted by e.g. on 2006-07-20 18:53:49