Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

摘要

It is shown that any generalized Jordan (triple-)derivation on a 2-torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2-torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.