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 by JLo)

After some more googling I came up with the following page:

http://eom.springer.de/I/i050210.htm .

It's third sentence says that Mal'tsev constructed an example of a cancellation ring that cannot be embedded in a division ring, and a reference is given to a 1937 paper by Mal'tsev.

Later I found the same reference to Mal'tsev (aka Malcev) in van der Waerden's Algebra.

A field of fractions is therefore not guaranteed to exist for every cancellation ring like it does for the commutative cancellation rings.

Posted by Richard on 2006-08-15 20:39:25