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.

Here is a truly ugly counter example of a monoid with left-cancellation property (LCP), but not RCP:
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Its members consist of all ordered pairs of non-negative integers. Here the multiplication operation:
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(a,b)*(x,y)=(a+x,y) for x unequal 0
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(a,b)*(x,y)=(a,b+y) for x equal 0
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Obviously (0,0) is the unit in this freak of nature. Remains to show associativity, LCP and absence of RCP. The proof is however as ugly and uninteresting as the monoid itself and is left to the reader (if there still is one for this thread...)

Posted by JLo on 2006-08-06 13:53:08