## Alternative loop rings.(English)Zbl 0537.17006

Let R be a commutative and associative ring without elements of additive order 2. Let L be a loop. The following conditions are equivalent: (i) The loop ring RL is alternative. (ii) If x,y,$$z\in L$$ are such that $$x.yz=xy.z$$ then $$a.bc=ab.c$$ whenever $$\{a,b,c\}=\{x,y,z\}$$ and if $$x.yz\neq xy.z$$ then $$xy.z=x.zy=y.xz.$$ (iii) L is an extra loop (i.e. L satisfies the identity $$(uv.w)u=u(v.wu)),\quad((x,y,z),x)=1$$ for all x,y,$$z\in L$$ and if a,b,$$c\in L$$ are such that $$a.bc\neq ab.c$$ then $$(a,b,c)=(a,b)=(b,c)$$ (here (d,e,f) and (d,e) denote the associator and the commutator, resp.).
Reviewer: T.Kepka

### MSC:

 17D05 Alternative rings 20N05 Loops, quasigroups

### Keywords:

alternative loop rings; Moufang loops; M(G,2) type; extra loop