Hypercentral units in alternative integral loop rings (Q1584624)

A Moufang loop \(L\) is called an RA loop if its loop ring \(RL\), for a commutative and associative ring \(R\) of characteristic \(\neq 2\), is alternative but not associative. For an RA loop \(L\), with the augmentation map \(\varepsilon : RL\rightarrow R\), the authors denote by \(U(RL)\) its loop of units, and by \(U_1(RL)=\{\alpha\in U(RL)|\varepsilon(\alpha)=1\}\), the loop of normalized units of \(RL\). It is proved that, for a finite RA loop \(L\), the hypercenter of \(U_1(ZL)\) coincides with its center, unless \(L\) is a Hamiltonian 2-loop, in which case \(L=U_1(ZL)\) coincides with its hypercenter.