When is (D, K) an S-accr pair?

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLet S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.Originality/valueMotivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

Amount algebras

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi [Formula: see text] conjecture, we prove that under some conditions, the formula [Formula: see text] holds for some amount [Formula: see text]-algebras [Formula: see text] and some ideals [Formula: see text] of [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text] that the ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing. A corollary to the mentioned formula is that if, for example, [Formula: see text] is a Prüfer domain or a torsion-free valuation ring and [Formula: see text] is a radical ideal of [Formula: see text], then [Formula: see text].

Prime avoidance property

Let [Formula: see text] be a commutative ring, we say that [Formula: see text] has prime avoidance property, if [Formula: see text] for an ideal [Formula: see text] of [Formula: see text], then there exists [Formula: see text] such that [Formula: see text]. We exactly determine when [Formula: see text] has prime avoidance property. In particular, if [Formula: see text] has prime avoidance property, then [Formula: see text] is compact. For certain classical rings we show the converse holds (such as Bezout rings, [Formula: see text]-domains, zero-dimensional rings and [Formula: see text]). We give an example of a compact set [Formula: see text], where [Formula: see text] is a Prufer domain, which has not prime avoidance property. Finally, we show that if [Formula: see text] are valuation domains for a field [Formula: see text] and [Formula: see text] for some [Formula: see text], then there exists [Formula: see text] such that [Formula: see text].

Divisibility Properties of the Semiring of Ideals of an Integral Domain

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

Pairs of domains where most of the intermediate domains are Prüfer

Let [Formula: see text] be an extension of integral domains. The ring [Formula: see text] is said to be maximal non-Prüfer subring of [Formula: see text] if [Formula: see text] is not a Prüfer domain, while each subring of [Formula: see text] properly containing [Formula: see text] is a Prüfer domain. Jaballah has characterized this kind of ring extensions in case [Formula: see text] is a field [A. Jaballah, Maximal non-Prüfer and maximal non-integrally closed subrings of a field, J. Algebra Appl. 11(5) (2012) 1250041, 18 pp.]. The aim of this paper is to deal with the case where [Formula: see text] is any integral domain which is not necessarily a field. Several examples are provided to illustrate our theory.

Quasi-strongly Gorenstein projective modules

In this paper, we introduce the class of quasi-strongly Gorenstein projective modules which is a particular subclass of the class of finitely generated Gorenstein projective modules. We also introduce and characterize quasi-strongly Gorenstein semihereditary rings. We call a quasi-strongly Gorenstein semihereditary domain a quasi-SG-Prüfer domain. A Noetherian quasi-SG-Prüfer domain is called a quasi-strongly Gorenstein Dedekind domain. Let [Formula: see text] be a field and [Formula: see text] be an indeterminate over [Formula: see text]. We prove that every ideal of the ring [Formula: see text] is strongly Gorenstein projective. We also show that every ideal of the ring [Formula: see text] (respectively, [Formula: see text]) is strongly Gorenstein projective. These domains are examples of quasi-strongly Gorenstein Dedekind domains.

Local types of Prüfer rings

Earlier papers by Glaz, Boynton, and the authors established hierarchies of properties for commutative rings in general which become equivalent to “Prüfer domain” for integral domains. Between Gaussian rings and Prüfer rings, these papers considered locally (respectively, maximally) Prüfer (respectively, strong Prüfer) rings. In this paper, we refine these hierarchies still further by considering the restriction of these local conditions to just the regular (respectively, semiregular) prime (respectively, maximal) ideals of the ring. In addition, we also consider these local conditions for [Formula: see text]-Prüfer rings. This refinement leads to a hierarchy of 21 conditions between locally strong Prüfer rings and Prüfer rings, of which 17 are inequivalent. We also determine the extent to which these conditions pass between a ring and its total quotient ring.