If we take a commutative ring $R$ whose prime ideal $I$ is considered for presence of zero divisors. I found the following reasoning:
$I$ is a prime ideal implies $I$ has no zero divisors.
FALSE: If $I$ is a prime ideal, this is equivalent to $R/I$ being an integral domain. But that concludes nothing about $I$. Thus $I$ may or may not have zero divisors.
The converse may not hold true in general, that is:
If an Ideal $I$ has no zero divisors, then it may not necessarily be a prime ideal.
I found the supporting counterexample:
$4Z$ is an ideal of $Z$ without zero divisors, but $4Z$ is not a prime ideal of $Z$ which can be checked easily.
Please verify if my arguments are valid. Thanks in advance.
As pointed out by awllower, your first claim is incorrect.
The ideal $\mathbb{C}\times\{0\}$ of $\mathbb{C}\times\mathbb{C}$ is prime, since the corresponding quotient is isomorphic to $\mathbb{C}$ , but $(1,0)\in\mathbb{C}\times\{0\}$ is a zero divisor.