Every maximal ideal is prime in a commutative ring with identity.
There were several posts on the site about analogues of the claim for rngs (rings with or without identity):
It looks like they do not provide the correct extension of the claim onto rngs.
We call an ideal maximal if it is a maximal proper ideal in the poset of ideals.
This notion assumes that the only "bigger" ideal for a maximal ideal is the principal ideal of units in a ring with identity.
It looks like the correct extension of the notion of a maximal ideal onto rngs is not a maximal proper ideal, but a maximal non-unit ideal (a maximal ideal in the poset of ideals that are not generated by units).
For example, the ideal $2 \mathbb Z$ is a maximal non-unit ideal in the ring with identity $\mathbb Z$, and it is prime;
the ideal $2 \mathbb Z$ is a maximal non-unit ideal in the ring without identity $2 \mathbb Z$, and it is prime.
In this case the claim for maximal ideals in rngs should be formulated in the following way:
every maximal non-unit ideal is prime in a commutative rng.
Is this correct?
Is there any use of the term "maximal non-unit ideal"?