Looking for example of a commutative non-unital ring in which every maximal ideal is a prime ideal

Question

Give example of a commutative non-unital ring in which every maximal ideal is a prime ideal.

The motivation for this question is : It is known that if $R$ is a commutative ring with identity $1 \ne 0$ and $M$ is a maximal ideal then $R/M$ is a field ; I have seen that if $R$ is a ring and $M$ is a maximal and prime ideal then also $R/M$ becomes a field ; also it is known that if $R$ is comm, ring with a multiplicative identity then every maximal ideal is prime ; hence arises the question of existence of a commutative ring without identity in which every maximal ideal is prime ...

Answer 1

In a ring without maximal ideals (which necessarily has no identity) the condition is vacuously satisfied. This page by Patrick Morandi is the most efficient way to get examples of these to you.

Here's another non-vacuous example. Let $F_2$ be the field of two elements, and consider the ideal $\oplus_{i=1}^\infty F_2\subseteq \prod_{i=1}^\infty F_2$. The right hand side is a ring with identity, of course, but the left hand side is a ring without identity. It's easy to see both rings are von Neumann regular rings. Let's denote the left hand ring by $R$.

If $M$ is a maximal ideal of $R$, then $M+(x)=R$ for any $x\notin M$. Since $R$ is von Neumann regular, $(x)=(e)$ for an idempotent element $e$. It easily follows that $R/M$ is a field with identity $e+M$, so $M$ is prime.