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.

Assume we have LCP (left-cancellation property). In order to prove RCP, say we have x≠y such that xb=yb. We must prove b=0:

xb=yb

follows

(x-y)b=0

follows

(x-y)(b+b)=(x-y)b

follows

b+b=b

follows

b=0

Posted by JLo on 2006-07-20 16:30:52