Let be a ring (not necessarily commutative) with .
Left zero divisor
is said to be a left zero divisor (LZD) if there is a in such that .
Right zero divisor
is said to be a right zero divisor (RZD) if there is a in such that .
is said to be a zero divisor (ZD) if it is both a left and a right zero divisor.
Left cancellation : .
Right cancellation : .
Prove that R has no non-trivial zero divisors if and only if both right and left cancellations hold.
Proof : Right cancellation holds will imply there are no nontrivial right zero divisors (). Similarly left cancellation holds will imply no nontrivial left zero divisors. So clearly no element can be a zero divisor except 0 (for it can’t even be a LZD or an RZD)
The other direction is more complicated.
- Assume is a nontrivial LZD and not a ZD (so x is not 0).
- So there is a such that .
- If also, then is an RZD and hence a ZD, a contradiction
- So .
- and , so yx is an LZD
- and , so yx is an RZD
- So is a nontrivial ( by Step 4) ZD.
So we have shown that is a nontrivial ZD.
Let us show left cancellation holds assuming no nontrivial ZD in R. If , this implies . If , we are done, else put . Thus is a nontrivial ZD, a contradiction.
Let us show right cancellation holds assuming no nontrivial ZD in R. If , this implies . If , we are done, else put and they are both not ZD as R has no nontrivial ZD. Thus is a nontrivial ZD, a contradiction.