Decompositions of rings (Q1907391)

The following theorem is proved: If an infinite associative division ring \(R\) is decomposed into a finite number of subsets \(R=A_1\cup\cdots\cup A_n\), then one can find an index \(m\) such that for the subset \(A=A_m\setminus\{0\}\) one has \(R=A^{-1}A-A^{-1}A+A^{-1}A-A^{-1}A=A^{-1}AA^{-1}A-A^{-1}AA^{-1}A\). For the proof are used properties of the right topological compact semigroup \(\beta (R\setminus\{0\})\) where \(\beta(R\setminus\{0\})\) is the Stone-Čech compactification of the discrete space \(R\setminus\{0\}\) [see \textit{N. Hindman}, Lect. Notes Math. 1401, 97-118 (1989; Zbl 0701.05060)].