Is there any non-monoid ring which has no maximal ideal?

Question

Is there any non-monoid ring which has no maximal ideal?

We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very easy Theorem there.

We say a ring $R$ is monoid if it has an multiplicative identity element, that if we denote this element with $1_{R}$ we should have: $\forall r\in R;\: r.1_{R}=1_{R}.r=r$

Answer 1

If $D$ is a valuation domain with unique maximal ideal $M$, then there are some conditions where $M$ is an example of a commutative rng with no maximal ideals.

As I remember it, one can choose a domain with a value group within $\Bbb{R}$ such that the group has no least positive element. Then, one can argue that the maximal ideal of that valuation domain $\{r\mid \nu(r)>0 \textrm{ or } r=0\}$ is a rng without maximal ideals.

Scanning the web, I think this pdf contains an argument of that sort.

Answer 2

Not every non-unital ring (or rng) has a maximal ideal. For example take $(\mathbb{Q},+)$ with trivial multiplication, i.e. $xy=0$ for all $x,y\in \mathbb{Q}$, then a maximal ideal is nothing more than a maximal subgroup. See this question why such a group does not exist.