The usual proof of "every prime ideal of a finite commutative ring with unity is maximal" is pretty straightforward.
$I$ is a prime ideal, so $R/I$ is a finite integral domain, so $R/I$ is a field, so $I$ is maximal.
But, how can we prove it without a quotient ring? or how can we prove it directly?
In order to illustrate what I mean by direct proof, let's look at the following example:
We can prove maximal ideals of a commutative ring with unity are prime as follows:
$I$ is maximal, so $R/I$ a is field, so $R/I$ is an integral domain, so $I$ is prime.
This approach is not direct, instead, you can check the direct approach here: https://math.stackexchange.com/q/68493
So, I want similar direct proof for my question.