Isomorphisms of incidence rings (Q788810)

For R a ring with identity and X a finite partially ordered set, denote by I(X,R) the incidence ring of R over X. We show that if R is an indecomposable semiperfect ring, then \(I(X,R)\cong I(Y,S)\) if and only if there is a ring T and (necessarily finite) partially ordered sets Z and W with \(R\cong I(Z,T)\), \(S\cong I(W,T)\), and \(X\times Z\cong Y\times W\). It follows that \(I(X,R)\cong I(X,S)\) implies \(R\cong S\) if R is semiperfect. Further, if R is a semiperfect ring that is not itself nontrivially an incidence ring, then any automorphism of I(X,R) is the composition of an inner automorphism, an automorphism induced by an order automorphism of X, and an automorphism induced by a family of additive maps from R to R that satisfy multiplication laws induced by the partial order. (Compare \textit{R. P. Stanley} [Bull. Am. Math. Soc. 76, 1236-1239 (1970; Zbl 0205.317)] which characterizes K-algebra automorphisms of I(X,K) for K a field.) Finally, we show that the incidence rings I(X,R) and I(X,S) have Morita duality if and only if R and S do. Hence I(X,R) is self-dual if and only if R is, answering a question left open by \textit{K. R. Fuller} and \textit{J. K. Haack} [in J. Pure Appl. Algebra 22, 113-119 (1981; Zbl 0466.16018)].