Abstract
PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D,K) to be anS-accr pair, whereDis an integral domain andKis a field which containsDas a subring andSis a multiplicatively closed subset ofD.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLetSbe a strongly multiplicatively closed subset of an integral domainDsuch that the ring of fractions ofDwith respect toSis not a field. Then it is shown that (D,K) is anS-accr pair if and only ifKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSinKis a one-dimensional Prüfer domain. LetD,S,Kbe as above. If each intermediate domain betweenDandKsatisfiesS-strong accr*, then it is shown thatKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSis a Dedekind domain; the separable degree ofKoverFis finite andKhas finite exponent overF, whereFis the quotient field ofD.Originality/valueMotivated by the work of some researchers onS-accr, the concept ofS-strong accr* is introduced and we determine some necessary conditions in order that (D,K) to be anS-strong accr* pair. This study helps us to understand the behaviour of the rings betweenDandK.
Highlights
The rings considered in this article are commutative with identity
Since the field K is an algebraic extension of the field F, using [12, Theorem 22.3], it can be shown as in the proof of [9, Proposition 2.1] that the integral closure of
This shows that (D, K) is an S-ACCRP. It follows from (1) 0 (2) of Theorem 2.12 that K is algebraic over D and the integral closure of SÀ1D in K is a onedimensional Pru€fer domain
Summary
The rings considered in this article are commutative with identity. The m.c. subsets considered in this article are assumed that they do not contain the zero element of the ring. Recall from [3, Definition 1] that M is said to satisfy (accr) (respectively, (accr*)) if for every submodule N of M and every finitely generated (respectively, principal) ideal B of R, the increasing sequence of residuals (N :MB) ⊆ (N :MB2) ⊆ (N :MB3) ⊆Á Á Á terminates. [3, 4] to modules and rings satisfying (accr). Various important properties of Noetherian modules and rings were generalized in Refs. [2] that M is called S-Noetherian if every submodule of M is an S-finite module. I am very much thankful to the referee and Dr Tariq Alfhadel for their suggestions and support
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