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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):

A maximal ideal is always a prime ideal?

and

Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal

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"?

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  • $\begingroup$ Rings which don't have identity don't have units, so defining maximality of ideals in rngs in terms of units is nonsensical. Those posts are completely correct as far as standard literature goes. $\endgroup$ – rschwieb Mar 13 at 13:29
  • $\begingroup$ @rschwieb I am using term "rng" for rings with or without identity. Is there an appropriate term for this type of objects? $\endgroup$ – Alex C Mar 13 at 13:36
  • $\begingroup$ The thing is that one may argue the definition of maximality must be strengthened to discuss them in rings without identity. For example, Jacobson's texts use maximal modular right ideals as the type of maximal right ideal that one should discuss, to eliminate bad cases like the ones given in the posts you linked. $\endgroup$ – rschwieb Mar 13 at 13:38
  • $\begingroup$ No, I would say it is fair to call "ring not necessarily with identity" a rng. What I'm saying is that if you define maximal ideals in terms of "not generated by units" it will have no meaning for the ones which do not have identity (but it will amount to the same thing for rings with identity, yes) so nothing is gained. $\endgroup$ – rschwieb Mar 13 at 13:39
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It looks like they do not provide the correct extension of the claim onto rngs.

It all looks standard to me.

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 is not generated by units). [...] 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.

If a ring does not have an identity, then it does not have units either (the definition of a unit requires the existence of an identity.) So, the proposed "better" definition for maximal ideals in rings does not have any meaning in a ring without identity.


One can argue, however, that the definition of a maximal ideal (for rings with identity) should be elaborated on to make it work in rings without identity.

One way to do this, as Jacobson did, is to additionally require the ideal to be modular. To accurately state it, he called a right ideal $T$ of $R$ modular if there exists an element $e\in R$ such that $ex=x$ for all $x\in T$. Put another way, there is an element that acts like a left identity on $T$. Notice how when a ring has identity, $e=1$ works for all right ideals maximal in the poset of proper right ideals, so they are all modular. This is a "good" extension of the "absolute" definition of maximal right ideals.

He used these ideals to characterize the Jacobson radical of rings without identity as the intersection of maximal modular right ideals (and not the "absolutely" maximal right ideals.)

In the most common example given in the posts you linked, the rng in question is $R=2\mathbb Z/4\mathbb Z$. Now, the zero ideal is certainly a maximal proper ideal in the ring, but it fails to be modular, as you can see. For this reason, $J(R)=R$, and not the zero ideal.

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  • $\begingroup$ Cannot we use the "maximal non-unit ideal" in multiple claims common for rings with or without identity, e.g. "a generator of a maximal non-unit principal ideal is irreducible in a commutative rng"? $\endgroup$ – Alex C Mar 13 at 13:52
  • $\begingroup$ @AlexC I can't make any sense of that comment. Can you please refine it to be clearer? $\endgroup$ – rschwieb Mar 13 at 14:09
  • $\begingroup$ I am trying to find common terminology for all the rings regardless of existence of identity. And it looks like replacing the word "proper" onto "non-unit" in the definition of a maximal ideal makes most of the claims that are true for rings with identity also true regardless of the identity. Does it make sense? $\endgroup$ – Alex C Mar 13 at 14:18
  • $\begingroup$ @AlexC Yes, thank you. The thing I don't understand at this instant is the distinction you are making between "proper" and "non-unit". As far as I know, "proper ideal=non-unit ideal". Can you explain how you believe they are different? $\endgroup$ – rschwieb Mar 13 at 14:27
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    $\begingroup$ Yes. And I am trying to go through the complications. At this point I think you gave me a clear and detail description of the problem. Thank you. I really appreciate your help. $\endgroup$ – Alex C Mar 13 at 19:53

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