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(2): Counter example for Monoids (actually it is valid) by JLo)
Apologies, JLo.
I did misread the rule. I agree that this has a unit
element and LCP and not RCP. I think that it is associative, and
therefore a monoid and a counterexample. Nice work!
re(3): Counter example for Monoids (actually it is valid)
