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( δ , 2 ) -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with nonzero identity, let ( ) be the set of all ideals of R and δ : ( ) ( ) an expansion of ideals of R defined by I δ ( I ) . We introduce the concept of ( δ , 2 ) -primary ideals in commutative rings. A proper ideal I of R is called a ( δ , 2 ) -primary ideal if whenever a , b R and a b I , then a 2 I or b 2 δ ( I ) . Our purpose is to extend the concept of 2 -ideals to ( δ , 2 ) -primary ideals of commutative rings. Then we investigate the basic properties of ( δ , 2 ) -primary ideals and also discuss the relations among ( δ , 2 ) -primary, δ -primary and...

A class of torsion-free abelian groups characterized by the ranks of their socles

Ulrich F. Albrecht, Anthony Giovannitti, H. Pat Goeters (2002)

Czechoslovak Mathematical Journal

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket R -module is R tensor a bracket group.

A simple characterization of principal ideal domains

Clifford S. Queen (1993)

Acta Arithmetica

1. Introduction. In this note we give necessary and sufficient conditions for an integral domain to be a principal ideal domain. Curiously, these conditions are similar to those that characterize Euclidean domains. In Section 2 we establish notation, discuss related results and prove our theorem. Finally, in Section 3 we give two nontrivial applications to real quadratic number fields.

Absolutely S-domains and pseudo-polynomial rings

Noomen Jarboui, Ihsen Yengui (2002)

Colloquium Mathematicae

A domain R is called an absolutely S-domain (for short, AS-domain) if each domain T such that R ⊆ T ⊆ qf(R) is an S-domain. We show that R is an AS-domain if and only if for each valuation overring V of R and each height one prime ideal q of V, the extension R/(q ∩ R) ⊆ V/q is algebraic. A Noetherian domain R is an AS-domain if and only if dim (R) ≤ 1. In Section 2, we study a class of R-subalgebras of R[X] which share many spectral properties with the polynomial ring R[X] and which we call pseudo-polynomial...

Almost perfect domains

S. Bazzoni, L. Salce (2003)

Colloquium Mathematicae

Commutative rings all of whose quotients over non-zero ideals are perfect rings are called almost perfect. Revisiting a paper by J. R. Smith on local domains with TTN, some basic results on these domains and their modules are obtained. Various examples of local almost perfect domains with different features are exhibited.

Almost Prüfer v-multiplication domains and the ring D + X D S [ X ]

Qing Li (2010)

Colloquium Mathematicae

This paper is a continuation of the investigation of almost Prüfer v-multiplication domains (APVMDs) begun by Li [Algebra Colloq., to appear]. We show that an integral domain D is an APVMD if and only if D is a locally APVMD and D is well behaved. We also prove that D is an APVMD if and only if the integral closure D̅ of D is a PVMD, D ⊆ D̅ is a root extension and D is t-linked under D̅. We introduce the notion of an almost t-splitting set. D ( S ) denotes the ring D + X D S [ X ] , where S is a multiplicatively closed...

Almost Q -rings

C. Jayaram (2004)

Archivum Mathematicum

In this paper we establish some new characterizations for Q -rings and Noetherian Q -rings.

Almost-free E(R)-algebras and E(A,R)-modules

Rüdiger Göbel, Lutz Strüngmann (2001)

Fundamenta Mathematicae

Let R be a unital commutative ring and A a unital R-algebra. We introduce the category of E(A,R)-modules which is a natural extension of the category of E-modules. The properties of E(A,R)-modules are studied; in particular we consider the subclass of E(R)-algebras. This subclass is of special interest since it coincides with the class of E-rings in the case R = ℤ. Assuming diamond ⋄, almost-free E(R)-algebras of cardinality κ are constructed for any regular non-weakly compact cardinal κ > ℵ...

Atomicity and the fixed divisor in certain pullback constructions

Jason Greene Boynton (2012)

Colloquium Mathematicae

Let D be an integral domain with field of fractions K. In this article, we use a certain pullback construction in the spirit of Int(E,D) that furnishes many examples of domains between D[x] and K[x] in which there are elements that do not admit a finite factorization into irreducible elements. We also define the notion of a fixed divisor for this pullback construction to characterize all of its irreducible elements and those nonzero nonunits that do admit a finite factorization into irreducibles....

Birings and plethories of integer-valued polynomials

Jesse Elliott (2010)

Actes des rencontres du CIRM

Let A and B be commutative rings with identity. An A - B -biring is an A -algebra S together with a lift of the functor Hom A ( S , - ) from A -algebras to sets to a functor from A -algebras to B -algebras. An A -plethory is a monoid object in the monoidal category, equipped with the composition product, of A - A -birings. The polynomial ring A [ X ] is an initial object in the category of such structures. The D -algebra Int ( D ) has such a structure if D = A is a domain such that the natural D -algebra homomorphism θ n : D i = 1 n Int ( D ) Int ( D n ) is an isomorphism for...

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