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A fundamental statement in stable homotopy theory/higher algebra/chromatic homotopy theory.
For any ring spectrum $R$ the kernel of the canonical morphism
to the MU-homology of $R$ consists of nilpotent elements.
This is due to Ethan Devinatz, Mike Hopkins, and Jeff Smith. See Lurie 10, theorem 3
The Nishida nilpotence theorem (Nishida 73) is the special case for the sphere spectrum, saying that all positive-degree elements in the stable homotopy groups of spheres are nilpotent.
This is indeed a special case. The MU-homology of the sphere spectrum is the Lazard ring and hence is torsion-free, whereas all positive-degree elements of the stable homotopy ring are torsion by the Serre finiteness theorem and therefore belong to the aforementioned kernel.
The nilpotence theorem implies that every ∞-field is an ∞-module over the Morava K-theory spectrum $K(n)$, for some $n$. See at ∞-field.
Goro Nishida, The nilpotency of elements of the stable homotopy groups of spheres, Journal of the Mathematical Society of Japan 25 (4): 707–732,
(1973) (euclid:euclid.jmsj/1240435467) doi:10.2969/jmsj/02540707, ISSN 0025-5645, MR 0341485.
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 25 The Nilpotence lemma (pdf)
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