Cancellation Laws in a Ring:
We say that cancellation laws hold in a ring , if
and where are in .
Thus in a ring with zero divisors, it is impossible to define a cancellation law.
Theorem: A ring has no divisor of zero if and only if the cancellation laws holds in .
Suppose that has no zero divisors. Let be any three elements of such that .
Thus the left cancellation law holds in . Similarly, it can be shown that right cancellation law also holds in .
Conversely, suppose that the cancellation laws hold in . Let and if possible let with then (because ).
Hence we get a contradiction to our assumption that and therefore the theorem is established.
Division Ring: A ring is called a division ring if its non-zero elements form a group under the operation of multiplication.
Pseudo Ring: A non-empty set with binary operations and satisfying all the postulates of a ring except right and left distribution laws, is called pseudo ring if