Units in integral loop rings
Let \(L\) be a loop and let \(L'\) and \(A(L)\) denote the subloops generated by all commutators and all associators respectively. Furthermore, denote by \(c(L)\) the set of elements of \(L\) that commute with every other element and by \(Z(L)\) the center of the loop; i.e., the set of elements in \(C(L)\) that also associate with any two other elements of \(L\), in any order. This paper considers the family \({\mathcal L}\) of all loops \(L\) such that \(L'\) and \(A(L)\) are contained in \(\{1,s\}\), a subloop of order 2, and \(Z(L)=C(L)\). This family includes RA loops and code loops and the authors show that it also contains many others that are not even Moufang loops. The main result is that if \(L\) is a diassociative loop in \({\mathcal L}\) then the units of the integral loop ring \(ZL\) are all trivial if and only if \(L\) is either commutative of exponent dividing 4 or 6 or a Hamiltonian 2-loop.
scientific article; zbMATH DE number 923701