Question from chapter on 'Integral Domains' from Gallian .which said an integral domain does not contains zero-divisor and is a commutative ring.
Does there exist any other commutative ring which is not a integral domain and also does not contain zero-divisor...
On
Yes. From my understanding, the ring $nZ$ for $n > 0$, with the usual multiplication and division, counts as a commutative ring without zero divisors but is not an integral domain because it doesn't contain unity (aka, multiplicative identity). I think your question stems from not noting that an integral domain is distinguished by containing unity.
Disclaimer: In this answer, the definition of a ring requires a multiplicative identity. If you use another definition of ring that does not make this requirement, then you might get a different answer - indeed, other people have provided such different answers.
We know that an integral domain doesn't contain (non-zero) zero-divisors because that (as well as the ring being non-zero itself) is the definition of an integral domain! So apart from the vacuous example of the zero ring, which has no non-zero zero-divisors but is not an integral domain by definition, there are no examples.