Bezout Domains and Rings with a Distributive Lattice of Right Ideals

1986 ◽  
Vol 38 (2) ◽  
pp. 286-303 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Distributive Lattice ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Integral Domains ◽  
Bezout Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains

It is the purpose of this paper to discuss a construction of right arithmetical (or right D-domains in [5]) domains, i.e., integral domains R for which the lattice of right ideals is distributive (see also [3]). Whereas the commutative rings in this class are precisely the Prüfer domains, not even right and left principal ideal domains are necessarily arithmetical. Among other things we show that a Bezout domain is right arithmetical if and only if all maximal right ideals are two-sided.Any right ideal of a right noetherian, right arithmetical domain is two-sided. This fact makes it possible to describe the semigroup of right ideals in such a ring in a satisfactory way; [3], [5].

On the Product of Ideals

1983 ◽  
Vol 26 (1) ◽  
pp. 106-114 ◽  
Author(s):  
David F. Anderson ◽  
David E. Dobbs
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Picard Group ◽  
Integral Domains ◽  
Valuation Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains

AbstractThis article introduces the concept of a condensed domain, that is, an integral domain R for which IJ = {ij: i ∊ I, j ∊ J} for all ideals I and J of R. This concept is used to characterize Bézout domains (resp., principal ideal domains; resp., valuation domains) in suitably larger classes of integral domains. The main technical results state that a condensed domain has trivial Picard group and, if quasilocal, has depth at most 1. Special attention is paid to the Noetherian case and related examples.

On the Criteria of D.D. Anderson for Invertible and Flat Ideals

1986 ◽  
Vol 29 (1) ◽  
pp. 25-32 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Nonzero Ideal ◽  
Finitely Generated ◽  
Integral Domains ◽  
Maximal Ideals ◽  
Dedekind Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains ◽  
Torsion Free

AbstractLet R be an integral domain. It is proved that if a nonzero ideal I of R can be generated by n < ∞ elements, then I is invertible (i.e., flat) if and only if I(∩ Rai) = ∩ Iai for all { a1, . . ., a n﹜ ⊂ I. The article's main focus is on torsion-free R-modules E which are LCM-stable in the sense that E(Ra ∩ Rb) = Ea ∩ Eb for all a, b ∈ R. By means of linear relations, LCM-stableness is shown to be equivalent to a weak aspect of flatness. Consequently, if each finitely generated ideal of R may be 2-generated, then each LCM-stable R-module is flat. Finally, LCM-stableness of maximal ideals serves to characterize Prüfer domains, Dedekind domains, principal ideal domains, and Bézout domains amongst suitably larger classes of integral domains.

scholarly journals Decomposition and classification of length functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1109-1129
Author(s):  
Dario Spirito
Keyword(s):  
Integral Domain ◽  
Integral Domains ◽  
Bijective Correspondence ◽  
Length Functions ◽  
Standard Representation ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

scholarly journals On the small and essential ideals in certain classes of rings

1989 ◽  
Vol 46 (2) ◽  
pp. 262-271 ◽  
Author(s):  
B. W. Green ◽  
L. van Wyk
Keyword(s):  
Commutative Rings ◽  
Matrix Rings ◽  
Small Ideal ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Complete Matrix ◽  
Structural Matrix

AbstractIt is well known that for a ring with identity the Brown-McCoy radical is the maximal small ideal. However, in certain subrings of complete matrix rings, which we call structural matrix rings, the maximal small and minimal essential ideals coincide.In this paper we characterize a class of commutative and a class of non-commutative rings for which this coincidence occurs, namely quotients of Prüfer domains and structural matrix rings over Brown-McCoy semisimple rings. A similarity between these two classes is obtained.

scholarly journals INTEGRAL DOMAINS WHOSE SIMPLE OVERRINGS ARE INTERSECTIONS OF LOCALIZATIONS

2005 ◽  
Vol 04 (02) ◽  
pp. 195-209 ◽  
Author(s):  
MARCO FONTANA ◽  
EVAN HOUSTON ◽  
THOMAS LUCAS
Keyword(s):  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

scholarly journals A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains

2019 ◽  
Vol 14 (2) ◽  
pp. 515-530
Author(s):  
Maria Francis ◽  
Thibaut Verron
Keyword(s):  
Principal Ideal ◽  
Gröbner Bases ◽  
General Setting ◽  
Polynomial Systems ◽  
Regular Sequence ◽  
Grobner Bases ◽  
Integral Domains ◽  
Unique Factorization Domain ◽  
Univariate Polynomials ◽  
Principal Ideal Domains

AbstractSignature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

scholarly journals DECIDABILITY OF THE THEORY OF MODULES OVER PRÜFER DOMAINS WITH INFINITE RESIDUE FIELDS

2018 ◽  
Vol 83 (04) ◽  
pp. 1391-1412 ◽  
Author(s):  
LORNA GREGORY ◽  
SONIA L’INNOCENTE ◽  
GENA PUNINSKI ◽  
CARLO TOFFALORI
Keyword(s):  
Prime Radical ◽  
Suitable Generalization ◽  
Prufer Domains ◽  
Residue Fields ◽  
Prüfer Domains ◽  
Bezout Domains

AbstractWe provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary.

On 1-absorbing prime ideals of commutative rings

2020 ◽  
pp. 2150175
Author(s):  
A. Yassine ◽  
M. J. Nikmehr ◽  
R. Nikandish
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Valuation Domains ◽  
Prufer Domains ◽  
Prüfer Domains

Let [Formula: see text] be a commutative ring with identity. In this paper, we introduce the concept of [Formula: see text]-absorbing prime ideals which is a generalization of prime ideals. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-absorbing prime if for all nonunit elements [Formula: see text] such that [Formula: see text], then either [Formula: see text] or [Formula: see text]. Some properties of [Formula: see text]-absorbing prime are studied. For instance, it is shown that if [Formula: see text] admits a [Formula: see text]-absorbing prime ideal that is not a prime ideal, then [Formula: see text] is a quasi–local ring. Among other things, it is proved that a proper ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing prime if and only if the inclusion [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text]. Also, [Formula: see text]-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

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