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(3): Another possibility? LCPRCP in monoids??? by JLo)
I haven't suceeded yet in either proving or disproving this for
monoids. I am about 99% certain that Richard's statement holds
for monoids with a finite number of elements. If there's a
counterexample, I think it will involve an infinite number of elements
in the monoid. No counterexample occurs to me, however. But
no proof occurs to me either.
re(4): Another possibility? LCPRCP in monoids???
