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??