Some remarks on simple alternative rings

K. A. Žhevlakov

Algebra i logika, Tome 6 (1967) no. 2, pp. 21-33.

Résumé

I. If $\mathcal{O}$ is a simple, commutative alternative ring then $\mathcal{O}$ is a field. II. Let $\mathcal{O}$ be a simple alternative ring of characteristic not $2,3$, then a) Jordan ring $\mathcal{O}^{(+)}$ is a simple ring. b ) If $J$ is an ideal of Malcev ring $\mathcal{O}^{(-)}$ then either $J$ contains $[\mathcal{O},\mathcal{O}]$ or $J$ is contained in center $Z$ of $\mathcal{O}^{(-)}$. In particular, if $\mathcal{O}^{(-)}$ is not Lie ring then $\mathcal{O}^{(-)}/Z$ a simple Malcev ring.