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

TL;DR

This paper extends the concept of dynamical IP$^{ ext{ extasteriskcentered}}$-sets to weak rings, demonstrating they contain complex algebraic configurations in a non-commutative setting.

Contribution

It establishes a non-commutative version of previous results, showing that dynamical IP$^{ ext{ extasteriskcentered}}$-sets in weak rings contain richer algebraic configurations.

Findings

Dynamical IP$^{ ext{ extasteriskcentered}}$-sets contain complex configurations in weak rings.

Extension of results to non-commutative algebraic structures.

Enrichment of combinatorial properties of IP$^{ ext{ extasteriskcentered}}$-sets.

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.