scholarly journals On 2-absorbing ideals of commutative semirings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050034
Author(s):  
H. Behzadipour ◽  
P. Nasehpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Minimal Prime ◽  
Unique Maximal Ideal

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if [Formula: see text] is a nonzero proper ideal of a subtractive valuation semiring [Formula: see text] then [Formula: see text] is a 2-absorbing ideal of [Formula: see text] if and only if [Formula: see text] or [Formula: see text] where [Formula: see text] is a prime ideal of [Formula: see text]. We also show that each 2-absorbing ideal of a subtractive semiring [Formula: see text] is prime if and only if the prime ideals of [Formula: see text] are comparable and if [Formula: see text] is a minimal prime over a 2-absorbing ideal [Formula: see text], then [Formula: see text], where [Formula: see text] is the unique maximal ideal of [Formula: see text].

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

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.

On n-absorbing and strongly n-absorbing ideals of amalgamation

2019 ◽  
Vol 19 (10) ◽  
pp. 2050199
Author(s):  
Mohammed Issoual ◽  
Najib Mahdou ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Ring Homomorphism ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula

Let [Formula: see text] be a commutative ring with [Formula: see text]. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal (respectively, a strongly n-absorbing ideal) of [Formula: see text] as in [D. F. Anderson and A. Badawi, On [Formula: see text]-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672] if [Formula: see text] and [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text] (respectively, if whenever [Formula: see text] for ideals [Formula: see text] of [Formula: see text], then the product of some [Formula: see text] of the [Formula: see text]s is contained in [Formula: see text]). The concept of [Formula: see text]-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of [Formula: see text] is a 1-absorbing ideal of [Formula: see text]). Let [Formula: see text] be a ring homomorphism and let [Formula: see text] be an ideal of [Formula: see text] This paper investigates the [Formula: see text]-absorbing and strongly [Formula: see text]-absorbing ideals in the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect [Formula: see text] denoted by [Formula: see text] The obtained results generate new original classes of [Formula: see text]-absorbing and strongly [Formula: see text]-absorbing ideals.

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.

On n-absorbing ideals and (m,n)-closed ideals in trivial ring extensions of commutative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950123 ◽  
Author(s):  
Ayman Badawi ◽  
Mohammed Issoual ◽  
Najib Mahdou
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
General Concept ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ring Extension ◽  
Text Recall ◽  
Ideal Formula

Let [Formula: see text] be a commutative ring with [Formula: see text]. Recall that a proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. A more general concept than 2-absorbing ideals is the concept of [Formula: see text]-absorbing ideals. Let [Formula: see text] be a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is called an n-absorbing ideal of [Formula: see text] if [Formula: see text] and [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. The concept of [Formula: see text]-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of [Formula: see text] is a 1-absorbing ideal of [Formula: see text]). Let [Formula: see text] and [Formula: see text] be integers with [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is called an [Formula: see text]-closed ideal of [Formula: see text] if whenever [Formula: see text] for some [Formula: see text] implies [Formula: see text]. Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] be an [Formula: see text]-module. In this paper, we study [Formula: see text]-absorbing ideals and [Formula: see text]-closed ideals in the trivial ring extension of [Formula: see text] by [Formula: see text] (or idealization of [Formula: see text] over [Formula: see text]) that is denoted by [Formula: see text].

scholarly journals NONCOMMUTATIVE HILBERT RINGS

2004 ◽  
Vol 03 (04) ◽  
pp. 437-443 ◽  
Author(s):  
ALGIRDAS KAUCIKAS ◽  
ROBERT WISBAUER
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
The Other ◽  
Prime Ideals ◽  
Natural Form ◽  
Noncommutative Rings ◽  
Ideal Formula ◽  
Contraction Property ◽  
Maximal Ideals

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n∈ℕ every maximal ideal [Formula: see text] contracts to a maximal ideal of R. This definition is also equivalent to [Formula: see text] being finitely generated as an [Formula: see text]-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

scholarly journals Characterizations of m-Normal Nearlattices in terms of Principal n-Ideals

2013 ◽  
Vol 38 ◽  
pp. 49-59
Author(s):  
MS Raihan
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Single Element ◽  
Finitely Generated ◽  
Fixed Element ◽  
Minimal Prime

A convex subnearlattice of a nearlattice S containing a fixed element n?S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal n-ideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In this paper, we include several characterizations of those Pn(S) which form m-normal nearlattices. We also show that Pn(S) is m-normal if and only if for any m+1 distinct minimal prime n-ideals Po,P1,…., Pm of S, Po ? … ? Pm = S. DOI: http://dx.doi.org/10.3329/rujs.v38i0.16548 Rajshahi University J. of Sci. 38, 49-59 (2010)

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.

MV-modules of fractions

2019 ◽  
Vol 19 (07) ◽  
pp. 2050131
Author(s):  
Hoda Banivaheb ◽  
Arsham Borumand Saeid
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Ideal Formula ◽  
One To One

In this paper, we study the notion of [Formula: see text]-modules of fractions with respect to a ⋅-prime ideal [Formula: see text]. Also the relation between the ideals of [Formula: see text]-module [Formula: see text] and the ideals of [Formula: see text]-module of fractions [Formula: see text] is studied and it is shown that there is a one to one correspondence between [Formula: see text]-prime [Formula: see text]-ideals of [Formula: see text]-module [Formula: see text] and [Formula: see text]-prime [Formula: see text]-ideals of [Formula: see text]-module of fractions [Formula: see text], where [Formula: see text] is the maximal localization of [Formula: see text] at a maximal ⋅-ideal [Formula: see text] of [Formula: see text] and [Formula: see text] is a ⋅-prime ideal of [Formula: see text] such that [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document