Generalized Differential Identities of (Semi–)Prime Rings

Let R be a semiprime ring with characteristic p ≥ 0 and RF be its left Martindale quotient ring. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \phi {\left( {X^{{\Delta _{j} }}_{i} } \right)} $$\end{document} is a reduced generalized differential identity for an essential ideal of R, then ϕ(Zije(Δj)) is a generalized polynomial identity for RF, where e(Δj) are idempotents in the extended centroid of R determined by Δj. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \phi {\left( {X^{{\Delta _{j} }}_{i} } \right)} $$\end{document} is a reduced generalized differential identity for a noncommutative Lie ideal of R, then ϕ(Zij) is a generalized polynomial identity for [R,R]. Moreover, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \phi {\left( {X^{{\Delta _{j} }}_{i} } \right)} $$\end{document} is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then ϕ(Zij is a generalized polynomial identity for Q.