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

Let $CAlg$ denote the $\infty$-categoory 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 $A \mapsto A^{red}:=A /nil(A)$.

We can define $$CAlg^{red} \hookrightarrow CAlg$$ as the $\infty$-cat. of $E_\infty$-rings whose underling ring is reduced.

Question: Does there exist a left adjoint? What about when we restrict to connective $E_\infty$-rings?

It seems too me that the approoach in HA,7.2.3 is somewhat relevant. I'd like to know if this this question is addressed somewhere in the literature.

[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 $A \mapsto A^{red}:=A /nil(A)$.

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.

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.