Extensions of zip rings

A ring R is called right zip provided that if the right annihilator r_R(X) of a subset X of R is zero, r_R(Y)=0 for a finite subset Y⊆X. Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; Characterize a ring R such that Mat_n(R) is right zip; When does R being a right zip ring imply R[G] being right zip when G is a finite group? In this note, we continue the study of the extensions of noncommutative zip rings based on Faith's questions.     

The unit graph of a left Artinian ring

Let R be a ring with non-zero identity and U(R) be the group of units of R. The unit graph of R, denoted by G(R), is a graph defined on the elements of R, and two distinct vertices r and s are adjacent if and only if r+s∈U(R). We investigate connectivity, diameter and the girth of the unit graph of a left Artinian ring. Also, by providing an algorithm, we determine when the unit graph of a finite ring is Hamiltonian.     

On zip and weak zip rings of skew generalized power series

In this paper we show under certain conditions that the skew generalized power series R[[S, w]] is a right zip (weak zip) ring if and only if R is a right zip (weak zip) ring.     

Principally Quasi-Baer skew Generalized Power Series modules

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M _R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M _R is p.q.-Baer if and only if the generalized power series module M[[S]] _{R[[S, ω]]} is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.     

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.     

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.     

Simple-Baer Rings and Minannihilator Modules

Let R be a ring. M is said to be a minannihilator left R-module if r _M l _R (I) = IM for any simple right ideal I of R. A right R-module N is called simple-flat if Nl _R (I) = l _N (I) for any simple right ideal I of R. R is said to be a left simple-Baer (resp., left simple-coherent) ring if the left annihilator of every simple right ideal is a direct summand of _R R (resp., finitely generated). We first obtain some properties of minannihilator and simple-flat modules. Then we characterize simple-coherent rings, simple-Baer rings, and universally mininjective rings using minannihilator and simple-flat modules.     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

On the genus of generalized unit and unitary Cayley graphs of a commutative ring

Let R be a commutative ring, U(R) be the set of all unit elements of R, G be a multiplicative subgroup of U(R) and S be a non-empty subset of G such that S ^?1={s ^?1:?s∈S}?S. In [16], K. Khashyarmanesh et al. defined a graph of R, denoted by Γ(R,G,S), which generalizes both unit and unitary Cayley graphs of R. In this paper, we derive several bounds for the genus of Γ(R,U(R),S). Moreover, we characterize all commutative Artinian rings R for which the genus of Γ(R,U(R),S) is one. This leads to the characterization of all commutative Artinian rings whose unit and unitary Cayley graphs have genus one.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

Minimal Overrings of an Integrally Closed Domain

Abstract

The main purpose of this paper is to characterize minimal overrings of an integrally closed domain R. We show that there exists a strong relationship between minimal overrings and the notion of ideal transforms. In particular, we prove that if T(M) = S(M) for each maximal ideal M, then there is a bijective correspondence between the set of invertible maximal ideals of R and the set of minimal overrings of R. This study enables us to produce several interesting applications concerning semi-local, Dedekind, Prüferian and Krull domains. Moreover, we investigate the spectrum of a minimal overring in comparison with the spectrum of R, and we determine whether the polynomial ring R[X ₁, X ₂,…, X _n ] has a minimal overring.     

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec _r (R) (resp. Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U _r (eR) = {P ? Spec _r (R)| e ? P}. Let  = ∪_{P?Prim r (R)} Spec _r ^P (R), where Spec _r ^P (R) = {Q ?Spec _r ^P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on  and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if  is a space with Zariski topology if and only if, for any Q ? , Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ? = Max _r (R) ∪  Prim _r (R) (resp.  = Prim _r (R)) there is an element e in R with e ² ? e ? J(R) such that U = U _r (eR) ∩  ? (resp. U = U _r (eR) ∩  ), where J(R) is the Jacobson of R. (3) Max _r (R) ∪  Prim _r (R) is connected if and only if Max _l (R) ∪  Prim _l (R) is connected if and only if Prim _r (R) is connected.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Strongly Duo Modules and Rings

An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M _R . Several equivalent conditions to being strongly duo are given. If M _R is strongly duo and reduced, then End _R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M _R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M _R is strongly duo if and only if either M ≈ R or M _R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R _R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring.     

Maurey-Rosenthal factorization of positive operators and convexity

We show that a Banach lattice X is r-convex, 1<r<∞, if and only if all positive operators T on X with values in some r-concave Köthe function spaces F(ν) (over measure spaces (Ω,ν)) factorize strongly through L_r(ν) (i.e., T=M_gR, where R is an operator from X to L_r(ν) and M_g a multiplication operator on L_r(ν) with values in F). This characterization of r-convexity motivates a Maurey-Rosenthal type factorization theory for positive operators acting between vector valued Köthe function spaces.     

F q [M 2], F q [GL 2], and F q [SL 2] as Quantized Hyperalgebras

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M _n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.     

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.     

On P-coherent endomorphism rings

A ring is called right P-coherent if every principal right ideal is finitely presented. Let M _R be a right R-module. We study the P-coherence of the endomorphism ring S of M _R . It is shown that S is a right P-coherent ring if and only if every endomorphism of M _R has a pseudokernel in add M _R ; S is a left P-coherent ring if and only if every endomorphism of M _R has a pseudocokernel in add M _R . Some applications are given.     

McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f(X)∈R[X] and m(X)∈M[X], f(X)m(X) = 0 implies there exists a nonzero r∈R (resp., m∈M) with rm(X) = 0 (resp., f(X)m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.