When is R[θ] integrally closed?

Let [Formula: see text] be an integrally closed domain with quotient field [Formula: see text] and [Formula: see text] be an element of an integral domain containing [Formula: see text] with [Formula: see text] integral over [Formula: see text]. Let [Formula: see text] be the minimal polynomial of [Formula: see text] over [Formula: see text] and [Formula: see text] be a maximal ideal of [Formula: see text]. Kummer proved that if [Formula: see text] is an integrally closed domain, then the maximal ideals of [Formula: see text] which lie over [Formula: see text] can be explicitly determined from the irreducible factors of [Formula: see text] modulo [Formula: see text]. In 1878, Dedekind gave a criterion known as Dedekind Criterion to be satisfied by [Formula: see text] for [Formula: see text] to be integrally closed in case [Formula: see text] is the localization [Formula: see text] of [Formula: see text] at a nonzero prime ideal [Formula: see text] of [Formula: see text]. Indeed he proved that if [Formula: see text] is the factorization of [Formula: see text] into irreducible polynomials modulo [Formula: see text] with [Formula: see text] monic, then [Formula: see text] is integrally closed if and only if for each [Formula: see text], either [Formula: see text] or [Formula: see text] does not divide [Formula: see text] modulo [Formula: see text], where [Formula: see text]. In 2006, a similar necessary and sufficient condition was given by Ershov for [Formula: see text] to be integrally closed when [Formula: see text] is the valuation ring of a Krull valuation of arbitrary rank (see [Comm. Algebra. 38 (2010) 684–696]). In this paper, we deal with the above problem for more general rings besides giving some equivalent versions of Dedekind Criterion. The well-known result of Uchida in this direction proved for Dedekind domains has also been deduced (cf. [Osaka J. Math. 14 (1977) 155–157]).