Products of generalized derivations on rings

TL;DR

This paper investigates conditions under which products of generalized derivations on rings are themselves generalized derivations, revealing that certain mixed products map into the ring's radical and establishing criteria for specific derivation combinations.

Contribution

It provides new conditions characterizing when the product of two generalized derivations is a generalized derivation, especially on prime rings with characteristic not two.

Findings

Products of generalized derivations can map into the radical of the algebra.

Necessary and sufficient conditions are established for certain derivation combinations to be generalized derivations.

Results apply specifically to prime rings with characteristic not equal to two.

Abstract

In this paper, we show that if the product $(D_1D_2, d_1d_2)$ of generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on an algebra $A$ is a generalized derivation, then $d_1D_2$ and $d_2D_1$ map $A$ into $\hbox{rad}(A)$. Also, for generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on a prime ring with characteristic different from two, we give necessary and sufficient conditions under which $(D_1^2+D_1D_2,d_1^2+d_1d_2)$ is a generalized derivation as well.