Prime Ideals In Commutative Rings Research Articles

On functional prime ideals in commutative rings

In this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a ∈ R and m ∈ M such that a ϕ ( m ) ∈ I , then a ∈ I or ϕ ( m ) ∈ I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ ∈ Hom R ( I , R ) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ ∈ Hom R ( I − 1 , R ) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided.

On n-1-absorbing prime ideals

In this paper, we introduce and study [Formula: see text]-1-absorbing prime ideals of commutative rings. Let [Formula: see text] be a ring and [Formula: see text] a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-1-absorbing prime ideal if whenever [Formula: see text] for some nonunits [Formula: see text] then either [Formula: see text] or [Formula: see text] It is obvious that 1-1-absorbing (2-1-absorbing) prime ideals are exactly prime (1-absorbing prime) ideals. Various examples and characterizations of [Formula: see text]-1-absorbing prime ideals are given.

On $n$-absorbing prime ideals of commutative rings

This paper investigates the class of rings in which every nn-absorbing ideal is a prime ideal, called nn-AB ring, where nn is a positive integer. We give a characterization of an nn-AB ring. Next, for a ring RR, we study the concept of Ω(R)={ωR(I);I is a proper ideal of R},Ω(R)={ωR(I);I is a proper ideal of R}, where ωR(I)=min{n;I is an n-absorbing ideal of R}ωR(I)=min{n;I is an n-absorbing ideal of R}. We show that if RR is an Artinian ring or a Prüfer domain, then Ω(R)∩NΩ(R)∩N does not have any gaps (i.e., whenever n∈Ω(R)n∈Ω(R) is a positive integer, then every positive integer below nn is also in Ω(R)Ω(R)). Furthermore, we investigate rings which satisfy property (**) (i.e., rings RR such that for each proper ideal II of RR with ωR(I)<∞ωR(I)<∞, $\omega_{R}(I)=\mid Min_R(I)\mid $ωR(I)=∣MinR(I)∣, where MinR(I)MinR(I) denotes the set of prime ideals of RR minimal over II). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern nn-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.

On 𝑆-weakly prime ideals of commutative rings

Abstract In this paper, we introduce and study the notion of an 𝑆-weakly prime ideal which is a generalization of a weakly prime ideal. Let 𝐴 be a commutative ring and 𝑆 a multiplicative set of 𝐴. We define a proper ideal 𝑃 of 𝐴 to be 𝑆-weakly prime if there exists s ∈ S s\in S such that, for all a , b ∈ A a,b\in A with 0 ≠ a ⁢ b ∈ P 0\neq ab\in P , then s ⁢ a ∈ P sa\in P or s ⁢ b ∈ P sb\in P . We describe the behavior of an 𝑆-weakly prime property across various ring-theoretic constructions such as direct product, homomorphic image, localization, trivial ring extensions and amalgamation rings.

On weakly 1-absorbing prime ideals

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $$A $$ be a commutative ring with a nonzero identity $$1\ne 0.$$ proper ideal $$P $$ of $$A $$ is said to be a weakly 1-absorbing prime ideal if for every nonunits $$x,y,z\in A $$ with $$0\ne xyz\in P, $$ then $$xy\in P$$ or $$z\in P. $$ In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in C(X), which is the ring of continuous functions of a topological space $$X. $$

On $$\phi $$-1-absorbing prime ideals

In this paper, we introduce $$\phi $$ -1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity $$1\ne 0$$ and $$\phi :\mathcal {I}(R)\rightarrow \mathcal {I}(R)\cup \{\emptyset \}$$ be a function where $$\mathcal {I}(R)$$ is the set of all ideals of R. A proper ideal I of R is called a $$\phi $$ -1-absorbing prime ideal if for each nonunits $$x,y,z\in R$$ with $$xyz\in I-\phi (I)$$ , then either $$xy\in I$$ or $$z\in I$$ . In addition to give many properties and characterizations of $$\phi $$ -1-absorbing prime ideals, we also determine rings in which every proper ideal is $$\phi $$ -1-absorbing prime.

On 1-absorbing prime ideals of commutative rings

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 two conjectures concerning -weakly prime ideals in commutative rings

Let R be a commutative ring with unity. A proper ideal P of R is called -weakly prime if implies for some where In 2014, Ebrahimpour [On generalizations of prime ideals (II). Comm. Algebra 42:3861–3875] posed two conjectures on -weakly prime ideals. In this paper, we prove them.

Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials

A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield \({\mathbb {T}}\), the semifield \(\mathbb {Z}_{\mathrm{max}}\) and the two element semifield \({\mathbb {B}}\). The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1. An improvement of a result from Bertram and Easton (Adv Math 308:36–82, 2017) is proven which can be regarded as a Nullstellensatz for tropical polynomials.

Two New Types of Rings Constructed from Quasiprime Ideals

Keigher showed that quasi‐prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi‐prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirrors of the corresponding results on von Neumann regular rings and principally flat rings (PF‐rings) in commutative rings, especially, for rings of positive characteristic.

SOME REMARKS ON PRINCIPAL PRIME IDEALS

AbstractIn this paper we investigate principal prime ideals in commutative rings. Among other things we characterize the principal prime ideals that are both minimal and maximal and characterize the maximal ideals of a polynomial ring that are principal. Our main result is that if (p) is a principal prime ideal of an atomic ring R, then ht(p)≤1.

A taxonomy of 2-primal rings

Various conditions on a noncommutative ring imply that it is 2-primal (i.e., the ring's prime radical coincides with the set of nilpotent elements of the ring). We will examine several such conditions and show that their known interdependencies are their only ones. Of particular interest will be the (PS I) condition on a ring (i.e., every factor ring modulo the right annihilator of a principal right ideal is 2-primal). We will see that even within a fairly narrow class of rings, (PS I) is a strictly stronger condition than 2-primal. We will show that the (PS I) condition is left-right asymmetric. We will also study the interplay between various types of semilocal rings and various types of 2-primal rings. The Köthe Conjecture will make a cameo appearance. In Section 6, we will examine subideals of prime ideals of commutative rings that are invariant under derivations.

Arrays of prime ideals in commutative rings

If R is a commutative ring, A and B ideals of R , and S and T multiplicative submonoids of R , we note an elementary necessary and sufficient condition for there to exist prime ideals P and Q in R such that P contains A and is disjoint from S , Q contains B and is disjoint from T , and P ⊆ Q . We then study conditions for the existence of larger families of prime ideals satisfying similar systems of relations. When the inclusion relations specified in the given system define a “tree order,” the necessary and sufficient conditions are quite tractable; otherwise, they are much less so. We apply these results to the case where R is a tensor product of two algebras over a field k , and end with some observations on the behavior of arrays of prime ideals in a k -algebra under base extension.

Intrinsic Spectral Topologies

ABSTRACT: Given a poset (X; ≤), any spectral topology with specialisation order ≤ lies between the upper topology and the Scott topology. Conditions are given under which these extreme topologies are indeed spectral. The results contribute to the understanding of speectral sets, that is, posets which arise as the prime ideals of commutative rings with 1, ordered by inclusion, or equivalently as the prime ideals of distributive lattices with 0 and 1, ordered by inclusion.

Lectures on Rings and Modules.

This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a particular emphasis on the concept of the complete ring of quotients. A survey of the fundamental concepts of algebras in the first chapter helps to make the treatment self-contained. The topics covered include selected results on Boolean and other commutative rings, the classical structure theory of associative rings, injective modules, and rings of quotients. The final chapter provides an introduction to homological algebra. Besides three appendices on further results, there is a six-page section of historical comments. Table of Contents: Fundamental Concepts of Algebra: 1.1 Rings and related algebraic systems; 1.2 Subrings, homomorphisms, ideals; 1.3 Modules, direct products, and direct sums; 1.4 Classical isomorphism theorems. Selected Topics on Commutative Rings: 2.1 Prime ideals in commutative rings; 2.2 Prime ideals in special commutative rings; 2.3 The complete ring of quotients of a commutative ring; 2.4 Rings of quotients of commutative semiprime rings; 2.5 Prime ideal spaces. Classical Theory of Associative Rings: 3.1 Primitive rings; 3.2 Radicals; 3.3 Completely reducible modules; 3.4 Completely reducible rings; 3.5 Artinian and Noetherian rings; 3.6 On lifting idempotents; 3.7 Local and semiperfect rings. Injectivity and Related Concepts: 4.1 Projective modules; 4.2 Injective modules; 4.3 The complete ring of quotients; 4.4 Rings of endomorphisms of injective modules; 4.5 Regular rings of quotients; 4.6 Classical rings of quotients; 4.7 The Faith-Utumi theorem. Introduction to Homological Algebra: 5.1 Tensor products of modules; 5.2 Hom and $\otimes$ as functors; 5.3 Exact sequences; 5.4 Flat modules; 5.5 Torsion and extension products. Appendixes; Comments; Bibliography; Index. Review from Zentralblatt Math: Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. Review from American Mathematical Monthly: An introduction to associative rings and modules which requires of the reader only the mathematical maturity which one would attain in a first-year graduate algebra [course]...in order to make the contents of the book as accessible as possible, the author develops all the fundamentals he will need. In addition to covering the basic topics...the author covers some topics not so readily available to the nonspecialist...the chapters are written to be as independent as possible...[which will be appreciated by] students making their first acquaintance with the subject...one of the most successful features of the book is that it can be read by graduate students with little or no help from a specialist. (CHEL/283.H)