Is there a definition of reduced $E_\infty$ ring?

Question

[Edit: I have completely changed the question in response to the replies given]

I am curious if there is well defined notion of reduced $E_\infty$-ring.

Let $CAlg$ denote the $\infty$-category of $E_\infty$-ring, $CAlg_1$, the one category of communicative rings. I would like to define the analog for reduced ring.

One categorically, $ CAlg_1^{red} \hookrightarrow CAlg_1$ admits a left adjoint $A \mapsto A^{red}:=A /nil(A)$.

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We can define $$CAlg^{red} \hookrightarrow CAlg$$ as the $\infty$-cat. of $E_\infty$-rings whose underling ring is reduced. Does there exist a left adjoint? As mentioned in comments by Marc, this is false.

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Question[Edit]: What should be the notion of $E_\infty$-ring? Harry in the comment says that this should be an ordinary reduced ring. I would appreciate if some explanation could make this precise.

Answer 1

As Harry points out, the usual approach (For example in Lurie's SAG) is to think of the map $R\to \pi_0(R)$ for a connective $E_\infty$ ring as some kind of quotient by a nil-ideal. This is justified by the fact that $R = \operatorname{lim}_n \tau_{\leq n} R$ is a limit of square-zero extensions $\tau_{\leq n} R\to \tau_{\leq n-1} R$. So if you work with connective $E_\infty$ rings, I think the only reasonable thing is to say that reduced ones are concentrated in degree $0$, and the reduction of $R$ is given by the (ordinary) reduction of $\pi_0(R)$.

This of course breaks down when allowing nonconnective rings. I think in that case there is no reasonable concept of reducedness, but of course it depends a lot on what you want to achieve with such a notion.