Two associativity theorems for alternative rings (Q1098927)

The author proves the following: (1) Let R be an alternative ring without nil ideals. If for any x,y,z\(\in R\) there exist natural numbers \(n=n(x,y,z)\), \(m=m(x,y,z)\), \(k=k(x,y,z)\), such that \((x^ n,y^ m,z^ k)=0\), then R is associative. (2) Let R be an alternative ring, and for any x,y,z\(\in R\) let there exist a polynomial \(p(t)=p_{x,y,z}(t)\) which has integer coefficients and is such that \((x^ 2p(x)-x,y,z)=0.\) Then R is associative. This second result was also announced by \textit{M. Slater} [Notices Am. Math. Soc. 23, No.1 (1976), A-3].