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: Another possibility? LCPRCP in groups??? by JLo)

I think that all groups have left and right cancellation properties, and that Richard's statement is true for groups (but not very interesting).

According to Wikipedia, the properties of all groups are:

• Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
• Neutral element: There is an element e in G such that for all a in G, e * a = a * e = a.
• Inverse element: For all a in G, there is an element b in G such that a * b = b * a = e, where e is the neutral element from the previous axiom.

We can then deduce left cancellation property as follows:

Let a' be the inverse element for a (one always exists).  
Then     ax = ay
Implies  a'(ax) = a'(ay)
Implies  (a'a)x = (a'a)y
Implies   ex = ey
Implies   x = y

A directly analgous proof deduces right cancellation property for all groups.

Edited on July 21, 2006, 8:22 am

Posted by Steve Herman on 2006-07-21 08:22:06