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.

Prove that
    having LCP (left-cancellation property)
is equivalent to
    having no zero-divisors
is equivalent to
    having RCP.

A ring has zero divisors if there are a ≠ 0 and b ≠ 0 such that ab=0.

The property in the middle is obviously symmetric, which closes the gap between LCP and RCP.

Posted by JLo on 2006-07-20 16:24:23