Dual coalgebras of algebras over commutative rings
Let \(R\) be a Noetherian ring and let \(A\) be an \(R\)-algebra, considered as a multiplicative monoid. The finite dual \(A^\circ\) is defined as the \(R\)-submodule of \(R^A\) consisting of \(R\)-valued representative functions on \(A\). The main result of this paper states that \(A^\circ\) is an \(R\)-coalgebra, provided that \(A^\circ\subseteq R^A\) is a pure \(R\)-submodule. This last condition is satisfied if \(R\) is a Noetherian hereditary ring. This result is applied by the two first authors and \textit{F. J. Lobillo} in their paper [J. Algebra 240, No. 1, 165-184 (2001; see the preceding review Zbl 0980.16028)]. A related result, for the case where \(R\) is a Dedekind domain, appears in the paper [Commun. Algebra 18, No. 10, 3209-3221 (1990; Zbl 0742.16018)] by \textit{C. Chen} and \textit{W. D. Nichols}.
scientific article; zbMATH DE number 1538660