We extend the notion of chain algebra, originally defined in \cite{GN} for finite distributive lattices, to that of finite pure posets. We show this algebra corresponds to the Ehrhart ring of a (0,1)-polytope, termed the chain polytope, and characterize the indecomposability of this polytope. Furthermore, we prove the normality of the chain algebra, describe its canonical module, and extend one of main results from \cite{GN} by computing its Krull dimension. For width-2 pure posets, we determine the algebra's regularity and conditions for it to be Gorenstein or nearly Gorenstein.