Skew incidence rings and the isomorphism problem

Let $X$ be a finite partially ordered set, $R$ an associative unital ring and $\sigma$ an endomorphism of $R$. We describe some properties of the skew incidence ring $I(X,R,\sigma)$ such as invertible elements, idempotents, the Jacobson radical and the center. Moreover, if the skew incidence rings $I(X,R,\sigma)$ and $I(Y,S,\tau)$ are isomorphic and the only idempotents of $R,S$ are the trivial ones, we show that the partially ordered sets $X$ and $Y$ are isomorphic.