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In this Article, we give a simple criterion for the regularity of atri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is abounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linearmapping, then $f$ is regular if and only if $hof$ is regular. We also shallgive some necessary and sufficient conditions such the fourth adjoint $D^{***}$of a tri-derivation $D$ is again tri-derivation.

Regularity of bounded tri-linear and the fourth adjiont of tri-derivation