Let us view some notions given in the summary. Each element \(\alpha\) of a finite dimensional algebra \(A\) over a perfect field \(\mathbb{F}\), has a unique additive Jordan decomposition \(\alpha= \alpha_s+\alpha_n\) (here \(\alpha_s\) is a semisimple element, \(\alpha_n\) is a nilpotent element, and \(\alpha_s\alpha_n=\alpha_n \alpha_s\) is fulfilled, too). “Semi simple” means here that \(\alpha_s\) has a minimal polynomial over \(\mathbb{F}\) with no repeated roots in the algebraic closure of \(\mathbb{F}\). If \(\alpha\) is a unit, then \(\alpha^{-1}_s\) exists and \(\alpha_u\) is by definition equal to \(1+ \alpha^{-1}_s\alpha_n\) satisfying \(\alpha_s\alpha_u=\alpha_u \alpha_s\). Then, \(\alpha=\alpha_s\alpha_u\) is the unique multiplicative Jordan decomposition of \(\alpha\). Next, assume \(A=\mathbb{Q}[g]\) and let \(\mathbb{Z}[g]\) be a subring of \(\mathbb{Q}[g]\), the integral group ring \(\mathbb{Z}[g]\). Each unit \(\alpha\) of \(\mathbb{Z}[g]\) has the unique multiplicative Jordan decomposition \(\alpha=\alpha_s \alpha_u\) in \(\mathbb{Q}[g]\). When it happens that for every unit \(\alpha\) of \(\mathbb{Z}[g]\) also the elements \(\alpha_s\) and \(\alpha_u\) are both contained in \(\mathbb{Z}[g]\), then, by definition, \(\mathbb{Z}[g]\) has the multiplicative Jordan decomposition property (MJD).
The author gives in the Introduction a survey of work done regarding the MJD in \(\mathbb{R}[g]\) for an integral domain \(\mathbb{R}\); see the list of references too. He works out explicitly that \(\mathbb{Z}[Q_8\times C_p]\) has the MJD. Here, \(Q_8\) is the quaternion group of order 8, \(C_p\) is a cyclic group of prime order \(p\), where also the multiplicative order of 2 modulo \(p\) is even together with \(p\) being congruent to 2 modulo 4. This “working out” is full of details, as provided in the paper.