The torsion product property in alternative algebras
Let \(R\) be an alternative ring with unity. Then \({\mathcal U}(R)\) denotes the set of units of \(R\), and \(T{\mathcal U}(R)\) is the set of the units of finite order in \(R\). The ring \(R\) has the torsion product property (briefly, ``has t.p.p.'') if the product of torsion units is again a torsion unit, i.e. if \(T{\mathcal U}(R)\) is a subloop of \({\mathcal U}(R)\). The authors pose the question under which conditions an alternative ring has the torsion product property. For alternative division rings and vector matrix algebras the authors establish the following results: (1) Let \(D\) be a division ring of characteristic \(p>0\). Then for any integer \(n>1\) the full matrix ring \(M_n(D)\) has t.p.p. if and only if \(D\) is a field which is an algebraic extension of its prime subfield. (2) Let \(F\) be a field of characteristic \(p>0\). Then Zorn's vector matrix algebra \({\mathfrak Z}(F)\) has t.p.p. if and only if \(F\) is algebraic over its prime field \(\mathcal P\). The main topic of this paper is the study of the torsion product property in alternative loop rings which are closely related to RA loops. An RA loop \(L\) is a loop with the property that for any commutative and associative ring \(R\) with unity the loop ring \(RL\) is alternative, but not associative. Such a loop \(L\) is necessarily a Moufang loop and thus diassociative. For RA loops the following hold: Assume \(L\) is an RA loop with torsion subloop \(T\). (a) Let \(F\) be a field of characteristic 0. Then the loop ring \(FL\) has t.p.p. if and only if \(T\) is an abelian group, for each \(t \in T\) and each \(x \in L\) there is a positive integer \(i\) such that \(xtx^{-1}=t^i\), and for each non central element \(t\in T\), \(F\) contains no root of unity whose order is the order of \(t\). Moreover, the torsion units of \(FL\) are contained in the units of \(FT\). (b) Let \(F\) be a field of characteristic \(p > 0\) with prime field \(\mathcal P\). Then \(FL\) has t.p.p. if and only if \(p=2\), or \(L=T\) and \(F\) is algebraic over \(\mathcal P\), or \(p\) is odd, \(T\) is an abelian group, and, if \(T\) is not central, then the algebraic closure, \(\overline{\mathcal P}\), of \(\mathcal P\) in \(F\) is finite and for all \(x \in L\) and all \(p'\)-elements \(a\in T\), one has \(xax^{-1}=a^{p^r}\) for some positive integer \(r\), which is a multiple of \([\overline{\mathcal P}:{\mathcal P}]\).
scientific article; zbMATH DE number 934242