Let $R$ be a ring spectrum. If $\pi\_\ast(R)$ is a graded field, then all module spectra over $R$ are free. But I don't believe the converse holds. How badly can it fail?


I'm assuming that $R$ is at least an $A\_3$ ring spectrum, and probably $A\_\infty$ (I'm also interested in more highly-structured cases); if the answer depends on just how highly structured $R$ is, I'd find that interesting.


Note that if $\alpha \in \pi\_\ast(R)$ is non-nilpotent, then the freeness of the localization $\alpha^{-1} R$ entails that $\alpha$ is a unit. So if all module spectra over $R$ are free, then $\pi\_\ast(R)$ is a local ring of dimension 1. But $\pi\_\ast(R)$ might have nonzero nilpotents as far as I can see.


**Question:** What is an example of a ring spectrum $R$ such that $\pi\_\ast(R)$ has nonzero nilpotents and yet every module spectrum over $R$ is free?



How weird can a ring spectrum be if all of its modules are free?
I think what you are looking for is the notion of a **semisimple** ring spectrum, as studied by [Hovey and Lockridge](https://eudml.org/doc/130465). For such a ring spectrum, $E$, every module spectrum is projective, i.e. is a retract of a coproduct of suspensions of $E$. The homotopy groups $E\_\*$ of such spectra are characterized in Theorem 1.2 of this paper: $E\_\* \cong R\_1 \times \dots \times R\_n$ where $R\_i$ is either a graded field $k$ or an exterior algebra $k[x]/(x^2)$ over a graded field with a unit in degree $3|x|+1$. Such rings $E\_\*$ are called **graded commutative $\Delta^1$-rings**, and clearly have plenty of non-zero nilpotent elements.


If $E$ is commutative, then $E$ is semisimple if and only if $E\_\*$ is a graded commutative $\Delta^1$-ring and for every factor ring of the form $k[x]/(x^2)$, we have $x\cdot \pi\_\*(C) \neq 0$ where $C$ is the cofiber of $x\cdot E$. 


I'll finish by mentioning classic work of [Hopkins and Smith](http://hopf.math.purdue.edu/Hopkins-SmithJH/nilpII.sidebyside.pdf), that the OP is surely aware of but future readers might not be, that defines the **fields** of stable homotopy theory as ring spectra $E$ such that $E\_\*X$ is a free $E\_\*$-module for all spectra $X$, and that characterizes fields as spectra having the homotopy type of a wedge of suspensions of Morava $K$-theory $K(n)$ for some fixed $n$ and prime $p$.


