  In this paper we study the structure and the algebraic varieties of
associative trialgebras. We provide a classification of n-dimensional
associative trialgebras for n $\leq$ 4. Using the classification result of
associative trialgebras, we describe the derivations and centroids of
low-dimensional associative trialgebras. We review some proprieties of the
centroids in light of associative trialgebras and and we calculate the
centroids of low-dimensional associative trialgebras.
