The nilpotence theorem for the algebraic $K$-theory of the sphere   spectrum
  We prove that in the graded commutative ring $K_{*}(\mathbb{S})$, all positive degree elements are multiplicatively nilpotent. The analogous statements also hold for $TC_{*}(\mathbb{S};\mathbb{Z}^{\wedge}_p)$ and $K_{*}(\mathbb{Z})$. 
Andrew J. Blumberg and Michael A. Mandell
math.KT math.AT