Normal subloops in the integral loop ring of an RA loop (Q2739249)

Let \(L\) be a loop which has an alternative loop ring over the ring \(\mathbb{Z}\) of rational integers. The loop of units in \(\mathbb{Z} L\), denoted by \(U(\mathbb{Z} L)\), is a Moufang loop which contains \(L\). A normal subloop \(N\) of \(U(\mathbb{Z} L)\) is called normal complement of \(L\) if \(L\cap N=\{1\}\) and \(U(\mathbb{Z} L)=\pm LN\).NEWLINENEWLINENEWLINEThe authors prove that: -- An RA loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring, -- if \(L\) and \(L_1\) are finite RA loops and \(\mathbb{Z} L\cong\mathbb{Z} L_1\), then \(L\cong L_1\), -- a finite RA loop \(L\) is normal in \(U(\mathbb{Z} L)\) iff \(U(\mathbb{Z} L)=\pm L\), -- a finite RA loop \(L\) is not normal in \(U(FL)\), where \(F\) is a field and \(U(FL)\) is not RA.