Factoring Ideals into Semiprime Ideals

1978 ◽  
Vol 30 (6) ◽  
pp. 1313-1318 ◽  
Author(s):  
N. H. Vaughan ◽  
R. W. Yeagy
Keyword(s):  
Integral Domain ◽  
Dedekind Domain ◽  
Prime Ideals ◽  
Semiprime Ideal ◽  
Dedekind Domains

Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.

On Representations of Orders Over Dedekind Domains

1960 ◽  
Vol 12 ◽  
pp. 107-125 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Integral Domain ◽  
Homological Algebra ◽  
Dedekind Domain ◽  
Main Concern ◽  
Direct Sums ◽  
Dedekind Domains ◽  
The Ideal

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

Primary Ideals in Prüfer Domains

1966 ◽  
Vol 18 ◽  
pp. 1024-1030 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Quotient Ring ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Prüfer Domain ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Primary Ideals

A Prüfer domain is an integral domainDwith the property that for every proper prime idealPofDthe quotient ringDPis a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).

scholarly journals Almost-Dedekind rings

1994 ◽  
Vol 36 (1) ◽  
pp. 131-134 ◽  
Author(s):  
E. W. Johnson
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Classical Result ◽  
Dedekind Domain ◽  
Ring Theory ◽  
Prime Ideals ◽  
Principal Ideals ◽  
Commutative Ring Theory ◽  
Lattice Of Ideals

Throughout we assume all rings are commutative with identity. We denote the lattice of ideals of a ringRbyL(R), and we denote byL(R)* the subposetL(R)−R.A classical result of commutative ring theory is the characterization of a Dedekind domain as an integral domainRin which every element ofL(R)* is a product of prime ideals (see Mori [5] for a history). This result has been generalized in a number of ways. In particular, rings which are not necessarily domains but which otherwise satisfy the hypotheses (i.e. general ZPI-rings) have been widely studied (see, for example, Gilmer [3]), as have rings in which only the principal ideals are assumed to satisfy the hypothesis (i.e. π-rings).

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

Radicals of PID's and Dedekind Domains

1972 ◽  
Vol 24 (4) ◽  
pp. 566-572 ◽  
Author(s):  
R. E. Propes
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Radical Class ◽  
Dedekind Domains ◽  
Radical Ideals ◽  
The One ◽  
Image Position ◽  
Principal Ideal Domains

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, ifthen T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

scholarly journals L-Fuzzy Semiprime Ideals of a Poset

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Berhanu Assaye Alaba ◽  
Derso Abeje Engidaw
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Semiprime Ideal

In this paper, we introduce the concept of L-fuzzy semiprime ideal in a general poset. Characterizations of L-fuzzy semiprime ideals in posets as well as characterizations of an L-fuzzy semiprime ideal to be L-fuzzy prime ideal are obtained. Also, L-fuzzy prime ideals in a poset are characterized.

scholarly journals Centralizer near-rings that are rings

1995 ◽  
Vol 59 (2) ◽  
pp. 173-183 ◽  
Author(s):  
Jutta Hausen ◽  
Johnny A. Johnson
Keyword(s):  
Endomorphism Ring ◽  
Sufficient Conditions ◽  
Dedekind Domain ◽  
Prime Spectrum ◽  
Dedekind Domains ◽  
Composition Of Functions

AbstractGiven an R-module M, the centralizer near-ring ℳR (M) is the set of all functions f: M → M with f(xr)= f(x)r for all x ∈ M and r∈R endowed with point-wise addition and composition of functions as multiplication. In general, ℳR(M) is not a ring but is a near-ring containing the endomorphism ring ER(M) of M. Necessary and/or sufficient conditions are derived for ℳR(M) to be a ring. For the case that R is a Dedekind domain, the R-modules M are characterized for which (i) ℳR(M) is a ring; and (ii)ℳR(M) = ER(M). It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.

scholarly journals RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

2012 ◽  
Vol 54 (3) ◽  
pp. 605-617 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
NOYAN ER ◽  
NİL ORHAN ERTAŞ
Keyword(s):  
Prime Ideals ◽  
Cyclic Module ◽  
Direct Sums ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Injective Modules ◽  
Dedekind Domains ◽  
Singular Ring

AbstractDedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

Arithmetic modules over generalized Dedekind domains

2020 ◽  
pp. 2250045
Author(s):  
Indah Emilia Wijayanti ◽  
Hidetoshi Marubayashi ◽  
Iwan Ernanto ◽  
Sutopo
Keyword(s):  
Free Module ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Dedekind Domains ◽  
Torsion Free Module ◽  
Polynomial Module ◽  
Torsion Free

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

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