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:

1. 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.
2. On the other hand, I find other sources such as this and this in the commutative framework which more or less say: to give a correct notion of ideal and of monomorphism is difficult, yet some attempts can be made.

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...