In a finite commutative ring , every prime ideal is maximal?

Question

I am stuck in a true/false question. It is

In a finite commutative ring, every prime ideal is maximal.

The answer says it's false.

Well what I can say is (Supposing the answer is right)

$(1)$ The ring can't be Integral domain since finite integral domain is a field.

$(2)$ There can't be unity in the ring since in that case the result would be true.(By the Theorem that if $R$ is a commutative ring with unity then an ideal $I$ is prime iff $R/I$ is Integral Domain)

$(3)$ All the elements are zero divisors since if there is at least one non- zero divisor, there will be a unity and so $(2)$ would follow.

So at the end, I am in search of a finite commutative with all elements as zero -divisors, having no unity and obviously a prime ideal in it which is not maximal.

What kind of strange looking ring is this (if possible) ? Any hints??

Answer 1

There is no counterexample, because even if the ring has no identity, the quotients by primes must have identity.

Every nonzero finite ring without zero divisors has a multiplicative identity, so the quotient would in fact be a finite domain with identity, and hence a field.

Answer 2

The answer is false. $I$ is prime means $R/I$ is a domain. Which implies $R/I$ is a field which implies that $I$ is maximal.