The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings) I'm trying to figure out what an ideal of a, say, dg-algebra (or, if you prefer, $\mathbb E_1$-ring) $R$ is in a homotopically relevant fashion, but I can't actually figure it out. I can assume that $R$ is concentrated in cohomologically nonpositive degrees (or homologically nonnegative degrees). I have stumbled upon a few possibilities:


*

*There is a notion of monomorphism in an $\infty$-category; hence, I would consider the derived category $\mathsf D(R)$ of $R$-dg-modules as an $\infty$-category and say that $I \to R$ is an "ideal" if it is a monomorphism according to that notion. This notion is also used in Spectral algebraic geometry (Remark C.2.3.4. page 1965) in the framework of Grothendieck prestable $\infty$-categories.

*On the other hand, I find other sources such as this and this. From the first one, I quote:


There are other things that are weird for commutative ring spectra. Quite often, we end up working with ideals in the graded commutative ring of homotopy groups, but as we saw above, this is not a suitable notion of ideal.There is a notion of an ideal in the context of (commutative) ring spectra [53] due to Jeff Smith, but still several algebraic constructions do not have an analogue in spectra.

Given that, I'm pretty confused. Perhaps the notion of monomorphism (1) is fine, but in the case of commutative ring spectra it does not work really well, hence the issues I found (2)? I've tried to skim through some literature on derived algebraic geometry, but still I couldn't find any satisfying answer...
 A: In (2), you linked to Mark Hovey's paper on Smith ideals, and mentioned "the commutative framework." But Hovey explicitly writes "we have not dealt with the commutative situation at all," so I don't know what you mean. However, if you do want a theory of commutative Smith ideals, you can find this in my first paper. Also, if you want a theory of Smith $O$-algebras, for an operad $O$ (e.g., $O = E_n$), then you can find this in a paper of mine with Donald Yau.
One of the crucial aspects of the story is that the algebraic structure on the morphism $f: I\to R$, viewed as an object in the arrow category, matches the algebraic structure on the cofiber of $f$ (at least, in stable settings where taking the cofiber makes sense). This is proven in section 4 of Hovey's paper, and in Theorem 4.4.1 of my paper with Donald Yau (with lots of examples occupying the rest of the paper). I'd like to write more, but have to run off to a Zoom meeting now for the rest of the day. Hopefully this observation gets you started. It's an important justification for the approach of both (1) and (2).
A: At least in commutative situations, I would argue that a good notion is simply an ideal in $H^0(R)$.
For example, the theory of local cohomology works just as well as it does for commutative rings, as long as you do it with respect to ideals in $H^0(R)$.
Similarly, you can take "derived quotients" with respect to a finite sequence of elements in $H^0(R)$, by taking the Koszul complex with respect to such a sequence.
See for example my very recent paper from last week:
"Koszul complexes over Cohen-Macaulay rings"
https://arxiv.org/abs/2005.10764
