Hoehnke radicals for right Lie algebras (Q1200942)

Let the assignment \(C\) designate an ideal \(C(L)\) to every nonassociative algebra \(L\) over a commutative ring \(F\), and define by induction \(T_ 0(L)=0\), \(T_{\alpha+1}(L)/T_ \alpha(L)=C(L/T_ \alpha(L))\), \(T_ \gamma(L)=\bigcup_{\beta<\gamma} T_ \beta(L)\) for limit ordinals \(\gamma\) and \(T(L)=\bigcup_{\alpha\geq 0} T_ \alpha(L)\). Further, let us consider condition \((S)\): there exists a set \(S\) of skew polynomials over \(F\) and for each \(f(x_ 1,\dots,x_ n)\in S\) there is an integer \(i\in\{1,\dots,n\}\) such that \(C(L)=\{a\in L\mid f(L,\dots,{\overset {i} a},\dots,L)=0\) for all \(f\in S\}\) holds for all \(L\). If the assignment \(C\) satisfies condition \((S)\), then the assignment \(T\) is a strongly hereditary idempotent Hoehnke radical, but in general not a Kurosh- Amitsur one. The author focusses the attention to right Lie algebras, i.e. algebras satisfying the identity \(x(yz)=(xy)z+y(xz)\). Right Lie algebras are generalizations of Lie algebras and every matrix algebra over a Lie algebra is the sum of isomorphic copies of right Lie algebras. For a certain set \(S\) of polynomials the author gets two different Hoehnke radicals \(T\) and \(T'\), and also intersection representations for these radicals. Considering right Lie algebras over a field \(F\) with \(\text{char }F\neq 2\) the right Lie algebras \(L\) with \(T(L)=0\) and \(T'(L)=0\) are described in terms of subdirect sums of matrix right Lie algebras over Lie algebras.