Prime radicals of graded \(\Omega\)-groups
A graded \(\Omega \)-group is an additive group \(A\) with a neutral element \(0\) graded by a group \(G\) with a system \(\Omega \) of \(n\)-ary algebraic operations such that for all \(\omega \in \Omega \) we have \((0,0,\dots,0) \omega =0\). The set \(\Omega \) is nonempty and it contains at least one \(n\)-ary operation such that \(n\geqslant 2\). The group \(A\) is a direct sum of normal subgroups \(A_{g}\), where \(g\in G\), which are called homogeneous components. For all \(a_{1}\in A_{g_{1}}\), \(a_{2}\in A_{g_{2}}\), \dots, \(a_{n}\in A_{g_{n}}\) and any \(n\)-ary operation \(\omega \in \Omega \) we have \(( a_{1},a_{2},\dots,a_{n}) \omega \in A_{g_{1}g_{2}\dots g_{n}}\). If, furthermore, for any finite set \(X\subseteq A\), for all \(n\geqslant 2\), \(a_{1},a_{2},\dots,a_{n}\in X\), \(\omega \in \Omega \), the set of elements \((a_{1},a_{2},\dots,a_{n}) \omega \) is finite, then the graded \( \Omega \)-group \(A\) is said to satisfy the finiteness condition. The elements of the set \(h( A) =\bigcup _{g\in G}A_{g}\) are called homogeneous elements of the \(\Omega \)-group \(A\). A nonempty subset \(I\) of a graded \(\Omega \)-group \(A\) is called a graded ideal if it satisfies the following three conditions: (1) \(I\) is a normal subgroup of the additive group \(A\); (2) for every \(n\)-ary operation \(\omega \in \Omega \) and any elements \(a\in I\) and \(x_{1},x_{2},\dots,x_{n}\in A\) we have \(-( x_{1},x_{2},\dots,x_{n}) \omega +(x_{1},\dots,x_{i-1},a+x_{i},x_{i+1},\dots,x_{n}) \omega \in I\), \( i=1,2,\dots,n\); and (3) if an element \(x\in I\) is the sum \( x=x_{g_{1}}+x_{g_{2}}+\dots+x_{g_{n}}\) of homogeneous elements \(x_{g_{i}}\in A_{g_{i}}\), \(g_{i}\in G\), then \(x_{g_{i}}\in I\) for all \(i=1,2,\dots,n\). In the paper the authors consider only graded ideals. If \(I\subseteq A\) is a graded ideal, then the relation \(a\thicksim b\Leftrightarrow a-b\in I\) is compatible with all operations and the factor set \(A/\!\!\thicksim \) is a graded \(\Omega \)-group which is called the \(\Omega\)-factor-group of the \(\Omega \)-group \(A\) by the ideal \(I\) and is denoted by \( A/I\). \(( a) \) denotes the graded ideal of a graded \(\Omega\)-group \(A\) generated by the element \(a\in A\). An element \(a\) of a graded \(\Omega\)-group \(A\) is called strictly Engel if any sequence of elements \( a_{0},a_{1},\dots\) of \(A\) satisfying the conditions \(a_{0}=a\) and \(a_{i+1}\in ( a_{i}) ^{2}\), where \(i=0,1,\dots\) is equal to \(0\) starting from some position. If \(I,J\subseteq A\) are graded \(\Omega \)-subgroups of a graded \(\Omega \)-group \(A\), then the mutual commutant \(\left[I,J\right] \) of these \(\Omega\)-subgroups is the \(\Omega \)-subgroup of \(A\) generated by the set of elements \((c_{1},c_{2},\dots,c_{n}) \omega \), where \(n\geqslant 2\), \(\omega \in \Omega \), \(c_{i}\) belong to \(I\) or \(J\) (for \(i=1,2,\dots,n\)) and, furthermore, some elements from \(I\) and from \(J\) necessarily occur among \( c_{i}\). The \(\Omega \)-subgroup \(\left[ I,I\right] \) is denoted by \(I^{2}\). A graded \(\Omega \)-group \(A\) is called solvable of degree \(k\) if there exists a natural number \(k\) such that \(A_{k}=0\) and \(A_{k-1}\neq 0\), where \( A_{0}=A\), \(A_{1}=\left[ A_{0},A_{0}\right] \), \dots , \(A_{i+1}=\left[ A_{i},A_{i}\right],\dots\). A graded \(\Omega \)-group \(A\) is called locally solvable if any of its finitely generated graded subgroups is solvable. A graded \(\Omega \)-group \(A\) is called weakly solvable if for any finite subset \(X\) of \(A\) there exists a natural number \(k\) such that \(X^{( k) }=0\), where \(X^{( 0) }=X\), \(X^{( k+1) }=\left\{ ( a_{1},a_{2},\dots,a_{n}) \omega :n\geqslant 2,\text{ } a_{1},a_{2},\dots,a_{n}\in X,\text{ }\omega \in \Omega \right\} \), \(k=0,1,\dots\). A graded \(\Omega \)-group \(A\) is called prime if for any two graded ideals \( I,J\subseteq A\) such that \(\left[ I,J\right] =0\) we have \(I=0\) or \(J=0\). Moreover, \(A\) is called semiprime if \(I^{2}=0\) implies \(I=0\). A graded ideal \(P\) of a graded \(\Omega \)-group \(A\) is called prime if the \(\Omega \)-factor-group \(A/P\) is prime. In this paper the authors define the graded prime radical \(P( A) \) of a graded \(\Omega \)-group \(A\) as the intersection of all graded prime ideals of the graded \(\Omega \)-group \(A\) and study properties of the radical. In particular, they show that the graded prime radical \(P(A)\) of a graded \(\Omega \)-group \(A\) coincides with the set of its strictly Engel elements. They show that a graded \(\Omega \)-group \(A\) is semiprime if and only if \(P( A) =0\). Moreover, they prove that the graded prime radical \(P( A) \) of a graded \(\Omega \)-group \(A\) with the finiteness condition is weakly solvable and \(P( A) \) is solvable if \(A\) is with the maximum condition on the graded ideals. They also show that if \(A\) is a graded \(\Omega \)-group with the finiteness condition, then the \(\Omega \)-group \(A/T( A) \) is a subdirect product of the graded prime \(\Omega \)-groups \(A_{\beta }\) such that \(T( A_{\beta }) =0\), where \(T( A_{\beta }) \) denotes the product of all graded weakly solvable ideals of \(A_{\beta }\). Similarly to the construction of the lower nil-radical of the associative algebras, the authors denote by \(\rho ( A) \) the sum of all graded ideals \(I\) in a graded \(\Omega \)-group \(A\) with the finiteness condition such that \(I^{2}=0\). Then, using transfinite induction, they define an ideal \(\rho ( \delta ) \) for every ordinal number \( \delta \) in the following way: (1) \(\rho ( 0) =0\); (2) let \(\rho ( \delta ) \) be defined for all \(\delta <\beta \), then let \(\rho ( \beta ) =\bigcup _{\gamma <\beta }\rho ( \gamma ) \) if \( \beta \) is a limit ordinal and, if \(\beta =\gamma +1\) is not a limit ordinal, then let \(\rho ( \beta ) \) be a graded ideal in \(A\) such that \(\rho ( \beta ) /\rho ( \gamma ) =\rho ( A/\rho ( \gamma ) ) \). They observe that there exists \( \beta \) such that \(\rho ( \beta ) =\rho ( \beta +1) \) and call \(\rho ( \beta ) \) the lower graded weakly solvable radical of the \text{graded }\(\Omega \)\text{-group }\(A\) with the finiteness condition. The authors show that the graded prime radical \(P( A) \) of a \text{graded }\(\Omega \)\text{-group }\(A\) with the finiteness condition coincides with the lower graded weakly solvable radical. In the remaining part of the paper the authors discuss examples of graded \( \Omega \)-groups which include associative algebras and graded associative algebras, Lie algebras and Lie superalgebras, conformal algebras, vertex algebras and groups.
solvable (locally solvable, weakly solvable) graded omega-group
upper (lower) graded weakly solvable radical of a graded omega-group
scientific article; zbMATH DE number 5358148
scientific article