In Nilpotence in Stable Homotopy Theory II, Hopkins–Smith define a field spectrum to be a ring spectrum $E$ such that $E_*X$ is a free $E_*$-module for all spectra $X$. They then show that any field spectrum $E$ has the homotopy type of a wedge of suspensions of a Morava K-theory $K(p,n)$ for some $n\in\mathbb{N}$ and some prime $p$. In this sense, the Morava K-theories are the prime fields in stable homotopy theory.

This is very different from the first definition of fields we usually see in commutative algebra, i.e. that $k$ is a field if $k^\times=k\setminus\{0\}$, however.

Question. Is it possible to define field spectra by means of some kind of invertibility condition, similar to the classical definition of fields?