The following theorem concerning integral group rings ZG was proved by Polcino Milies and Sehgal in [4] (see also Jespers, Juriaans, de Miranda, and Rogerio [3]).

Theorem. Let Φ(G) denote the FC subgroup of a group G and let T = TΦ(G) be the torsion subgroup of Φ(G). Then every central unit u of ZG can be written in the form u = wg with w ∈ ZT and g ∈ Φ(G). Recently Dokuchaev, Polcino Milies, and Sehgal [1] characterized precisely

those groups G such that ZG has only trivial central units, using the above theorem to reduce the general case to a case with more manageable groups (the case where G is finite had been settled earlier by Ritter and Sehgal [6]). Clearly the decomposition u = wg is not unique - in fact, u = (wt−1)(tg)

for any t ∈ T . In this paper we investigate the following question. Question. Is it possible to arrange the decomposition u = wg in such a way that w and g are both central units in ZG?