Isomorphisms of incidence rings
type: Item
instance of: scholarly article
title: Isomorphisms of incidence rings
zbMATH Open document ID: 0532.16017
author: Joel K. Haack
published in: Illinois Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 3843950
zbMATH Keywords: finite partially ordered set
zbMATH Keywords: incidence ring
zbMATH Keywords: indecomposable semiperfect ring
zbMATH Keywords: inner automorphism
zbMATH Keywords: order automorphism
zbMATH Keywords: Morita duality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3843950