Cancellation Laws in a Ring

Cancellation Laws in a Ring:
We say that cancellation laws hold in a ring R, if
ab = bc\,\left( {a \ne 0}  \right)\,\, \Rightarrow \,b = c and ba = ca\,\,\left( {a \ne 0} \right)\,\,\,  \Rightarrow b = c where a,b,c are in R.
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 R.

Suppose that R has no zero divisors. Let a,b,c be any three elements of R such that a \ne 0,\,\,ab = ac.

\begin{gathered} ab = ac\, \Rightarrow ab - ac = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  \Rightarrow a\left( {b - c} \right) = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  \Rightarrow b - c = 0\,\,\,\,\,\,\,\,\,\,\,\,\left[  {{\text{because}}\,R\,{\text{is}}\,{\text{without}}\,{\text{zero}}\,{\text{divisor}}\,{\text{and}}\,a  \ne 0} \right] \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  \Rightarrow b = c \\ \end{gathered}

Thus the left cancellation law holds in R. Similarly, it can be shown that right cancellation law also holds in R.
Conversely, suppose that the cancellation laws hold in R. Let a,b  \in R and if possible let ab = 0 with a \ne 0,\,b \ne 0 then ab = a \cdot 0 (because a \cdot 0 = 0).
Since a \ne 0,\,\,ab  = a \cdot 0 \Rightarrow b = 0
Hence we get a contradiction to our assumption that b \ne 0 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 R with binary operations  + and  \times satisfying all the postulates of a ring except right and left distribution laws, is called pseudo ring if

\left(  {a + b} \right) \cdot \left( {c + d} \right) = a \cdot c + a \cdot d + b \cdot  c + b \cdot d

for all a,b,c,d  \in R