2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172293When $n$ is odd, a cohomology of type Hochschild for $n$-ary partially associative algebras has been defined in Gnedbaye's thesis. Unfortunately, the cohomology definition is not valid when $n$ is even. This fact is found again in the computations of the $n$-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for $n$-ary partially associative algebras. First by reducing the space of cochains, secondly by using a graded version. Next we compute the free $n$-ary algebra, giving a basis of this algebra. At last we extend the notion of coalgebras to $n$-ary algebras.23 pagesRings and Algebrastext