Nilpotence theorem

In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum $$\mathrm{MU}$$. More precisely, it states that for any ring spectrum $R$, the kernel of the map $\pi_\ast R \to \mathrm{MU}_\ast(R)$ consists of nilpotent elements. It was conjectured by and proved by.

Nishida's theorem
showed that elements of positive degree of the homotopy groups of spheres are nilpotent. This is a special case of the nilpotence theorem.