Combined Algebraic Properties in Gaussian and Quaternion Ring

Abstract

It is known that for an IP^{*} set A in (\mathbb{N},+) and a sequence \left\langle x_{n}\right\rangle _{n=1}^{\infty} in \mathbb{N}, there exists a sum subsystem \left\langle y_{n}\right\rangle _{n=1}^{\infty} of \left\langle x_{n}\right\rangle _{n=1}^{\infty} such that FS\left(\left\langle y_{n}\right\rangle _{n=1}^{\infty}\right)\cup FP\left(\left\langle y_{n}\right\rangle _{n=1}^{\infty}\right)\subseteq A. Similar types of results have also been proved for central^{*} sets and C^{*}-sets where the sequences have been considered from the class of minimal sequences and almost minimal sequences. In this present work, our aim to establish the similar type of results for the ring of Gaussian integers and the ring of integer quaternions.