Let \(A\) be an algebra (not necessarily associative) over a field \(\Phi\) of characteristic different from 2 and 3. Denoting the product in \(A\) by juxtaposition one forms the bracket \([a,b]=ab-ba\) and the symmetrized product \(a\bullet b=(ab+ba)/2\), \(a\), \(b\in A\). Let also \((a,b,c)=(ab)c-a(bc)\) be the usual associator of \(a\), \(b\), \(c\in A\). For \(\gamma\), \(\delta\in\Phi\) such that \(\gamma^2-\delta^2+\delta-1=0\), the variety of \((\gamma,\delta)\)-algebras is defined by the identities \((x,x,x)=0\), \((x,y,z)+\gamma(y,z,x)+\delta(z,x,y)=0\), \((x,y,z)-\gamma(x,z,y)+(1-\delta)(y,z,x)=0\). It is easy to see that \((-1,1)\)-algebras are exactly right-alternative algebras satisfying the Jacobi identity.
If \(A\) is a right-alternative or Jordan algebra then \(a\in A\), \(a\ne 0\) is an absolute zero divisor if \(a^2=0\) and \(aAa=0\). If \(A\) has no absolute zero divisors it is nondegenerate, otherwise it is called degenerate. A prime degenerate algebra is called a monster. The existence of monster algebras was established by the author of the present paper in [S. V. Pchelintsev, Algebra Logic 24, 441–454 (1985; Zbl 0603.17014); translation from Algebra Logika 24, No. 6, 674–695 (1985)], and in [S. V. Pchelintsev, Math. USSR, Izv. 28, 79–98 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 1, 79–100 (1986; Zbl 0609.17008)].
The paper under review studies primeness properties of right-alternative and of Jordan algebras and their relations. Recall that \(A\) is strongly \((-1,1)\)-algebra (St-algebra for short) if it is prime nonassociative \((-1,1)\)-algebra; they satisfy also the identity \([[x,y],z]=0\). One of the main results in the paper under review states that if \(A\) is St-algebra, generated by a Jordan subalgebra \(J\) then \(J\) is prime as well. Moreover it is proved that the converse also holds, in a sense. Call \(A\) an St-envelope of \(J\). Then if \(J\) is prime then there exists an appropriate St-envelope which is prime. (The algebra \(J\) is called then St-special.) Furthermore the author proves that ideals of prime St-special Jordan algebras are also prime. A corollary to this theorem is that metaideals of a Jordan monster are prime algebras.
Finally the author studies prime algebras in characteristic \(p>3\). He proves that prime nil-algebras related to the monster algebras, satisfy the identity \(x^p=0\). Moreover if \(A\) is strongly \((-1,1)\)-algebra then its associator ideal satisfies that same identity.