Dynamical IP${}^{\star }$-sets in weak rings

Abstract

V. Bergelson and N. Hindman proved that IP${}^{\star }$- sets contain all possible finite sums and products of a sum subsystem of any sequence in $\mathbb{N}$. In a recent work the second author of this article has proved that a stronger result holds for dynamical IP${}^{\star }$- sets. In this article we will establish a non-commutative version of this result. We will prove that a richer configuration is contained in dynamical IP${}^{\star }$- sets in weak rings.