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).

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.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

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.

scholarly journals A Remark on Coherent Overrings

1978 ◽  
Vol 21 (3) ◽  
pp. 373-375 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
General Theorem ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Quotient Field ◽  
Valuation Rings ◽  
Finitely Presented

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

Graded 1-absorbing prime ideals of graded commutative rings

2021 ◽  
Author(s):  
Mohammed Issoual
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Graded Ring

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

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