Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R, it is shown that the local duality establishes a bijection between the preinjective left R-modules and the preprojective right R-modules, and any preinjective left R-module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R-modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361-376] for hereditary rings.     

Locally Artinian Serial Rings

Abstract

In this paper we extend a well-known characterization of unital Artinian serial rings to the analagous class of rings with local units. In particular we characterize locally Artinian serial rings as those in which every finitely generated module or every finitely cogenerated module is uniserial. Additionally, we show that the entire characterization does not extend, as there exist locally Artinian serial rings that have non-serial modules.     

Noetherian Rings whose Modules are Prime Serial

A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (?-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\), and we say that M is prime serial (?-serial, for short) if it is a direct sum of ?-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ?-serial?” and “Which rings have the property that every finitely generated module is ?-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring.     

Weakly regular rings with ACC on annihilators and maximality of strongly prime ideals of weakly regular rings

Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal.     

Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

Let M₁,…,M_n be right modules over a ring R. Suppose that the endomorphism ring of each module M_i has at most two maximal right ideals. Is it true that every direct summand of M₁⊕?⊕M_n is a direct sum of modules whose endomorphism rings also have at most two maximal right ideals? We show that the answer is negative in general, but affirmative under further hypotheses. The endomorphism ring of uniserial modules, that is, the modules whose lattice of submodules is linearly ordered under inclusion, always has at most two maximal right ideals, and Pavel P?íhoda showed in 2004 that the answer to our question is affirmative for direct sums of finitely many uniserial modules.     

Rugged modules: The opposite of flatness

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M⁺ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.     

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.     

Divisibility classes are seldom closed under flat covers

We show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring $R/sR$ satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

ON SOME PROPERTIES OF IF RINGS

Abstract

We show that left IF rings (rings such that every injective left module is flat) have certain regular-like properties. For instance, we prove that every left IF reduced ring is strongly regular. We also give characterizations of (left and right) IF rings. In particular, we show that a ring R is IF if and only if every finitely generated left (and right) ideal is the annihilator of a finite subset of R.     

Commutative rings whose proper ideals are direct sums of uniform modules

An 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 ? of the form ? = 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.     

Unique addition modules

LetV be a module over a ring R. We define V to be a unique addition module (UAM) if it is not possible to change the addition of V without changing the action of R on V. If R is a domain such that 1 is not the only unit in R and V is a torsion-free i2-module, then we show that V is a UAM if and only if V has rank 1 (or if V={0}). We also classify finitely generated unique addition modules over commutative Artinian rings.     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.     

IG-projective modules

This paper concerns finitely generated modules over Artin algebras. We introduce the notion of an IG-projective module and use this to prove that if, over such an algebra R$R$, each simple module is strongly Gorenstein projective, then any indecomposable R$R$-module is either projective or simple. We also prove that if R$R$ is local and the simple module is IG-projective, then 1-self-orthogonal modules are projective.     

Rings whose cyclic modules are lifting and ⊕-supplemented

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules.     

On Finitely Annihilated Modules and Artinian Rings

Let R be a ring. An R-module M is finitely annihilated if the annihilator of M is the annihilator of a finite subset of M. It is proved that if R has right socle S then the ring R/S is right Artinian if and only if every singular right R-module is finitely annihilated. Moreover, a right Noetherian ring R is right Artinian if and only if every uniform right R-module is finitely annihilated. In addition, a (right and left) Noetherian ring is (right and left) Artinian if and only if every injective right R-module is finitely annihilated. This paper will form part of the Ph.D. thesis at the University of Glasgow of the second author. He would like to thank the EPSRC for their financial support     

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.