Upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) in terms of the order of the ring and/or its number of units?

Question

Is there any known upper bound for the no. of maximal ideals of a finite commutative ring ( with unity ) of order $n$ , in terms of $n$ and /or in terms of the no. of units of the ring ? Say , does there exist a finite commutative ring ( with unity ) of order $n$ such that it has more than $n$ maximal ideals ? Any reference or link is highly appreciated . Thanks in advance

Answer 1

If $R$ is a finite commutative ring with maximal ideals $I_1,\dots,I_k$ then by the Chinese Remainder Theorem there is a surjective homomorphism $$R\to R/I_1\times\dots\times R/I_k,$$ and so $\vert R\vert\geq\prod_{i=1}^k\vert R/I_i\vert$.

Since each $R/I_i$ has at least two elements, this gives $\log_2(\vert R\vert)$ as an upper bound for the number of maximal ideals, which is attained if $R$ is a direct product of copies of $\mathbb{F}_2$.