Regularity of Bounded Tri-Linear Maps and the Fourth Adjoint of a Tri-Derivation

In this Article, we give a simple criterion for the regularity of a trilinear mapping. We provide if f : X × Y × Z → W is a bounded trilinear mapping and h : W → S is a bounded linear mapping, then f is regular if and only if hof is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint D∗∗∗∗ of a triderivation D is again triderivation.