I see two "obvious" classes of finite $E_\infty$ ring spectra $R$:

• $R = \Sigma^\infty_+ (S^1)^{\times n}$

• $R = D\Sigma^\infty_+ X$ ($X$ a finite space)

Questions:

1. Are there any others?

2. In all the above examples, the unit map $\mathbb S \to R$ splits off. Is this always the case?

3. In the second class of examples, I believe all elements of $\pi_\ast R$ not in the image of the unit $\pi_\ast \mathbb S \to \pi_\ast R$ are nilpotent. How generally is this true? Is it true for all examples not in the first class of examples?

4. How does the answer change if we localize at a prime, or perform some more drastic localization?

I see two "obvious" classes of nonzero finite $E_\infty$ ring spectra $R$:

• $R = \Sigma^\infty_+ (S^1)^{\times n}$

• $R = D\Sigma^\infty_+ X$ ($X$ a finite space)

Questions:

1. Are there any others?

2. In all the above examples, the unit map $\mathbb S \to R$ splits off. Is this always the case?

3. In the second class of examples, I believe all elements of $\pi_\ast R$ not in the image of the unit $\pi_\ast \mathbb S \to \pi_\ast R$ are nilpotent. How generally is this true? Is it true for all examples not in the first class of examples?

4. How does the answer change if we localize at a prime, or perform some more drastic localization?