Proper Ideal Research Articles

Generalizations of 2-absorbing primal ideals in commutative rings

Let $R$ be a commutative ring with unity $(1\not=0)$. A proper ideal of $R$ is an ideal $I$ of $R$ such that $I\not=R$. Let $\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}$ be any function, where $\mathfrak{I}(R)$ denotes the set of all proper ideals of $R$. In this paper we introduce the concept of a $\phi$-2-absorbing primal ideal which is a generalization of a $\phi$-primal ideal. An element $a\in R$ is defined to be $\phi$-2-absorbing prime to $I$ if for any $r,s,t\in R$ with $rsta\in I\setminus\phi(I)$, then $rs\in I$ or $rt\in I$ or $st\in I$. An element $a\in R$ is not $\phi$-2-absorbing prime to $I$ if there exist $r,s,t\in R$, with $rsta\in I\setminus\phi(I)$, such that $rs, rt, st\in R\setminus I$. We denote by $\nu_\phi(I)$ the set of all elements in $R$ that are not $\phi$-2-absorbing prime to $I$. We define a proper ideal $I$ of $R$ to be a $\phi$-2-absorbing primal if the set $\nu_\phi(I)\cup\phi(I)$ forms an ideal of $R$. Many results concerning $\phi$-2-absorbing primal ideals and examples of $\phi$-2-absorbing primal ideals are given.

Commutative rings whose proper ideals are pure-semisimple

Recall that an R-module M is pure-semisimple if every module in the category is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that “a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring”. Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is “whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?” The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple.

A property of ideals of jets of functions vanishing on a set

For a set E\subset\mathbb{R}^n that contains the origin, we consider I^m(E) – the set set of all m^{\text{th}} degree Taylor approximations (at the origin) of C^m functions on \mathbb{R}^n that vanish on E . This set is a proper ideal in \mathcal{P}^m(\mathbb{R}^n) – the ring of all m^{\text{th}} degree Taylor approximations of C^m functions on \mathbb{R}^n . Which ideals in \mathcal{P}^m(\mathbb{R}^n) arise as I^m(E) for some E ? In this paper we introduce the notion of a closed ideal in \mathcal{P}^m(\mathbb{R}^n) , and prove that any ideal of the form I^m(E) is closed. We do not know whether in general any closed proper ideal is of the form I^m(E) for some E , however we prove in a subsequent paper that all closed proper ideals in \mathcal{P}^m(\mathbb{R}^n) arise as I^m(E) when m+n\leq5 .

Structure of rings with commutative factor rings for some ideals contained in their centers

This article concerns commutative factor rings for ideals contained in the center. A ring $R$ is called CIFC if $R/I$ is commutative for some proper ideal $I$ of $R$ with $I\subseteq Z(R)$, where $Z(R)$ is the center of $R$. We prove that (i) for a CIFC ring $R$, $W(R)$ contains all nilpotent elements in $R$ (hence Köthe's conjecture holds for $R$) and $R/W(R)$ is a commutative reduced ring; (ii) $R$ is strongly bounded if $R/N_*(R)$ is commutative and $0\neq N_*(R)\subseteq Z(R)$, where $W(R)$ (resp., $N_*(R)$) is the Wedderburn (resp., prime) radical of $R$. We provide plenty of interesting examples that answer the questions raised in relation to the condition that $R/I$ is commutative and $I\subseteq Z(R)$. In addition, we study the structure of rings whose factor rings modulo nonzero proper ideals are commutative; such rings are called FC. We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.

Choosing between incompatible ideals

Suppose I and J are proper ideals on some set X. We say that I and J are incompatible if I∪J does not generate a proper ideal. Equivalently, I and J are incompatible if there is some A⊆X such that A∈I and X∖A∈J. If some B⊆X is either in I∖J or in J∖I, then we say that Bchooses between I and J.We consider the following Ramsey-theoretic problem: Given several pairs (I1,J1),(I2,J2),…,(Ik,Jk) of incompatible ideals on a set X, find some A⊆X that chooses between as many of these pairs of ideals as possible. The main theorem is that for every n∈N, there is some I(n)∈N such that given at least I(n) pairs of incompatible ideals on any set X, there is some A⊆X choosing between at least n of them.This theorem is proved in two main steps. The first step is to identify a (purely finitary) problem in extremal combinatorics, and to show that our problem concerning ideals is equivalent to this combinatorial problem. The second step is to analyze the combinatorial problem in order to show that the number I(n) described above exists, and to put bounds on it. We show 12nlog2n−O(n)<I(n)<nlnn+O(n).The upper bound is proved by considering a different but closely related combinatorial problem involving hypergraphs, which may be of independent interest. We also investigate some applications of this theorem to a problem concerning conditionally convergent series.

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.

Prereductions of ideals in local rings

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then:There is a natural one-to-one correspondence between the set of the equivalence classes of the type-prereductions of I and the set of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring (t an indeterminate) such that and the Krull dimension of is equal to one). is the set of maximal elements in for all positive integers n, and, for each for all large integers m.A complete description is given of the prereductions and the elements in for each ideal I that is generated by analytically independent elements.If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets and and for all positive integer n.

Products of ideals in Leavitt path algebras

Ideals in Leavitt path algebras share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (e.g., prime, prime-power, primary, irreducible, semiprime, quasi-primary) has proved fruitful, we conduct an analogous investigation for Leavitt path algebras. Specifically, we classify the proper ideals in these rings that admit factorizations into products of each of the above types. We also classify the Leavitt path algebras where every proper ideal admits a factorization of these sorts, as well as those Leavitt path algebras where every proper ideal is of one of those types.

On 1-absorbing primary ideals of commutative rings

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

Some results on a subgraph of the intersection graph of ideals of a commutative ring

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. Let us denote the collection of all proper ideals of $R$ by $mathbb{I}(R)$ and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$. With $R$, we associate an undirected graph denoted by $g(R)$, whose vertex set is $mathbb{I}(R)^{*}$ and distinct vertices $I_{1}, I_{2}$ are adjacent in $g(R)$ if and only if $I_{1}cap I_{2}neq I_{1}I_{2}$. The aim of this article is to study the interplay between the graph-theoretic properties of $g(R)$ and the ring-theoretic properties of $R$.

Serial factorizations of right ideals

In a Dedekind domain D, every non-zero proper ideal A factors as a product A=P1t1⋯Pktk of powers of distinct prime ideals Pi. For a Dedekind domain D, the D-modules D/Piti are uniserial. We extend this property studying suitable factorizations A=A1…An of a right ideal A of an arbitrary ring R as a product of proper right ideals A1,…,An with all the modules R/Ai uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of h-local Prüfer domains and that of semirigid commutative GCD domains.

A characterization of commutative rings whose maximal ideal spectrum is Noetherian

An ideal [Formula: see text] of a ring [Formula: see text] is called pseudo-irreducible if [Formula: see text] cannot be written as an intersection of two comaximal proper ideals of [Formula: see text]. In this paper, it is shown that the maximal spectrum of [Formula: see text] is Noetherian if and only if every proper ideal of [Formula: see text] can be expressed as a finite intersection of pseudo-irreducible ideals. Using a result of Hochster, we characterize all [Formula: see text] quasi-compact Noetherian topological spaces.

Commutative rings whose proper ideals are direct sums of uniform modules

ABSTRACTAn interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ℳ of the form ℳ = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.

Commutative Rings whose Proper Ideals are Serial

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal ${\mathcal {M}}$ such that there exist a uniserial module U and a semisimple module T with ${\mathcal {M}}=U\oplus T$ . Moreover, in the latter case, every proper ideal of R is isomorphic to $U^{\prime }\oplus T^{\prime }$ , for some $U^{\prime }\leq U$ and $T^{\prime }\leq T$ . Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal ${\mathcal {M}}$ containing a set of elements {w 1,…,w n } such that ${\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}$ with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results.

Decomposition of ideals into pseudo-irreducible ideals

ABSTRACTA proper ideal of a commutative ring is called pseudo-irreducible if it cannot be written as a product of two comaximal proper ideals. In this paper, we give a necessary and sufficient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal pseudo-irreducible ideals. Examples of such rings include Laskerian rings, or more generally J-Noetherian rings and ZD-rings. We study when certain classes of rings satisfy this condition.

The annihilating-ideal graph of commutative semigroups

In this paper, we associate an undirected graph AG(S), the annihilating-ideal graph, to a commutative semigroup S. This graph has vertex set A⁎(S)=A(S)∖{(0)}, where A(S) is the set of proper ideals of S with nonzero annihilator. Two distinct vertices I,J∈A⁎(S) are defined to be adjacent in AG(S) if and only if IJ=(0), the zero ideal. Conditions are given to ensure a finite graph. Semigroups for which each nonzero, proper ideal of S is an element of A⁎(S) are characterized. Connections are drawn between AG(S) and Γ(S), the well-known zero-divisor graph, and the connectivity, diameter, and girth of AG(S) are described. Semigroups S for which AG(S) is a complete or star graph are characterized. Finally, it is proven that the chromatic number is equal to the clique number of the annihilating ideal graph for each reduced semigroup and null semigroup. Upper and lower bounds for χ(AG(S)) are given for a general commutative semigroup.

Commutative rings whose proper ideals are direct sum of completely cyclic modules

By two results of Köthe and Cohen–Kaplansky we obtain that “a commutative ring [Formula: see text] has the property that every [Formula: see text]-module is a direct sum of (completely) cyclic modules if and only if [Formula: see text] is an Artinian principal ideal ring” (an [Formula: see text]-module [Formula: see text] is called completely cyclic if each submodule of [Formula: see text] is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring [Formula: see text] is a direct sum of completely cyclic [Formula: see text]-modules if and only if [Formula: see text] is a principal ideal ring or [Formula: see text] is a local ring with maximal ideal [Formula: see text] such that there is an index set [Formula: see text] and a set of elements [Formula: see text] such that [Formula: see text] with each [Formula: see text] a simple [Formula: see text]-module and [Formula: see text] a principal ideal ring.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Cardinalities of proper ideals in some lattices of strengthenings of the intuitionistic propositional logic

We prove that each proper ideal in the lattice of axiomatic, resp. standard strengthenings of the intuitionistic propositional logic is of cardinality 2ℵ0. But, each proper ideal in the lattice of structural strengthenings of the intuitionistic propositional logic is of cardinality 22ℵ0. As a corollary we have that each of these three lattices has no atoms.

Closed ideals in the group algebra 𝐿¹(𝐺)∩𝐿²(𝐺)

0. Introduction. In the following, G will denote a locally compact abelian topological group with character group For 1 < p < oo, LP(G) is the Banach space of all complex-valued functions whose pth powers are Haar integrable over (LP(G) is often written LP when the group G is obvious from the context.) The linear space L'(G) n L2(G) (denoted LI n L2) is normed in such away that, under convolution as multiplication, it is a commutative Banach algebra (?2). It is also proved in ?2 that it is regular, semi-simple and that its regular maximal ideal space is 6. Itis shown (?3) that the abstract Silov theorem [8, p. 86] holds for LI n L2. The standard proof of this theorem in L'(G) seems to depend upon the uniform boundedness of the approximate identity. A novel aspect of the LI nl L2 case is that a similar proof is obtained despite the fact that every approximate identity in L' rn L2 is unbounded. An important but unsolved problem of harmonic analysis is the classification of the closed ideals in L'(G). Using the additional structure supplied by LI n L2 it is to be expected that more precise results can be obtained about the closed ideals in LI n L2. If G and G are both locally compact metric abelian groups, examples of the more precise results that can be obtained are: (a) If I is a closed proper ideal in LI n L2, then there exists an x E I such that the hull of x and the hull of I coincide except for a set of measure zero (Theorem 7.2). (b) For every closed invariant proper subspace N c L2(G), N n L' = k(h(N n LI)) (Corollary 2 of Theorem 7.4). This permits a new characterization of the kernel of E for a class of perfect sets E c G. (A. Denjoy terms these sets epais en lui-meme in Lepons sur le calcul des coefficients d'une serie trigonometrique, Paris, 1941, 2ieme Partie, p. 100.) (c) The set f of all closed proper ideals in LI n L2 which are