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In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if $${\text {Tor}}_{1}^{R}(A, R/I)= 0$$ for any maximal left ideal I of R. A right module B is said to be max-cotorsion if $${\text {Ext}}^{1}_{R}(A,B)=0$$ for any max-flat right module A. We characterize some classes of rings such as perfect rings, max-injective rings, SF rings and max-hereditary rings by max-flat and max-cotorsion modules. We prove that every right module has a max-flat cover and max-cotorsion envelope. We show that a left perfect right max-injective ring R is QF if and only if maximal right ideals of R are finitely generated. The max-flat dimensions of modules and rings are studied in terms of right derived functors of $$-\otimes -$$ . Finally, we study the modules that are injective and flat relative to s-pure exact sequences.

For a right module [Formula: see text], we prove that [Formula: see text] is simple-injective if and only if [Formula: see text] is min-[Formula: see text]-injective for every cyclic right module [Formula: see text]. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is [Formula: see text] if and only if simple-injective right modules are projective. For a commutative Noetherian ring [Formula: see text], we prove that every finitely generated simple-injective [Formula: see text]-module is projective if and only if [Formula: see text], where [Formula: see text] is [Formula: see text] and [Formula: see text] is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.

Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right $H$-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

Let R be a ring with unity and let $$M_R$$ and $$_RN$$ be right and left modules,respectively. The module $$M_R$$ is said to be absolutely $$_RN$$ -pure if $$M \otimes N \rightarrow L \otimes N$$ is amonomorphism for every extension $$L_R$$ of $$M_R$$ . For a module $$M_R$$ , the subpurity domain of $$M_R$$ is defined to be the collection of all modules $$_RN$$ , such that $$M_R$$ is absolutely $$_RN$$ -pure. Clearly, $$M_R$$ is absolutely $$_RF$$ -pure for every flat module $$_RF$$ and that $$M_R$$ is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, $$M_R$$ is said to be a test for flatness by subpurity (or t.f.b.s. for short) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. We characterize the structure of t.f.b.s. modules over commutative hereditary Noetherian rings. We prove that a module M is t.f.b.s. over a commutative hereditary Noetherian ring if and only if M / Z(M) is t.f.b.s. if and only if $${\text {Hom}}(M/Z(M), S) \ne 0$$ for each singular simple module S. Prufer domains are characterized as those domains all of whose nonzero finitely generated ideals are t.f.b.s.

Weakening the notion of [Formula: see text]-projectivity, a right [Formula: see text]-module [Formula: see text] is called max-projective provided that each homomorphism [Formula: see text], where [Formula: see text] is any maximal right ideal, factors through the canonical projection [Formula: see text]. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are [Formula: see text]-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring [Formula: see text], we prove that injective modules are [Formula: see text]-projective if and only if [Formula: see text], where [Formula: see text] is [Formula: see text] and [Formula: see text] is a small ring. If [Formula: see text] is right hereditary and right Noetherian then, injective right modules are max-projective if and only if [Formula: see text], where [Formula: see text] is a semisimple Artinian and [Formula: see text] is a right small ring. If [Formula: see text] is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective.

Lopez-Permouth, Sergio/0000-0002-7376-2167, WOS: 000428834300002, As a mean to provide intrinsic characterizations of poor modules, the notion of a pauper module is introduced. A module is a pauper if it is poor and has no proper poor direct summand We show that not all rings have pauper modules and explore conditions for their existence. In addition, we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity: one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule. The second condition holds for the ring of integers and is just as significant as the first one for Noetherian rings since, in that context, modules having poor pure submodules must themselves be poor. It is shown that the existence of paupers is equivalent to the Noetherian condition for rings with no middle class As indecomposable poor modules are pauper, we study rings with no indecomposable right middle class (i.e. the ring whose indecomposable right modules are pauper or injective). We show that semiartinian V-rings satisfy this property and also that a commutative Noetherian ring R has no indecomposable middle class if and only if R is the direct product of finitely many fields and at most one ring of composition length 2. Structure theorems are also provided for rings without indecomposable middle class when the rings are Artinian serial or right Artinian. Rings for which not having an indecomposable middle class suffices not to have a middle class include commutative Noetherian and Artinian serial rings. The structure of poor modules is completely determined over commutative hereditary Noetherian rings. Pauper Abelian groups with torsion-free rank one are fully characterized. (C) 2018 Elsevier Inc. All rights reserved.

Lopez-Permouth, Sergio/0000-0002-7376-2167, WOS: 000418083100023, 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=SxT, 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.

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism Y → M is neat in Y, i.e., the induced map Hom(S, Y) → Hom(S, M) is surjective for any simple right R-module S. Neat-flat right R-modules are projective if and only if R is a right (Formula presented.) -CS ring. Every cyclic neat-flat right R-module is projective if and only if R is right CS and right C-ring. It is shown that, over a commutative Noetherian ring R, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only if R ≅ A × B, wherein A is a QF-ring and B is hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only if R ≅ A × B, wherein A is a QF-ring and B is Artinian with J 2(B) = 0., Scientific and Technical Research Council of Turkey

In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to $\oplus_{p \in P} \Z_p$, where $P$ is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of $U^{(\mathbb{N})}$, where $U$ ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group $M$, it is shown that $M$ can not be torsion, and each $p$-primary component of $M$ is unbounded. Finally, we show that there are pi-poor groups which are not poor, and vise versa., The necessity part of Theorem 3.1. is proved without using basic subgroups

All modules considered in this note are over associative com- mutative rings with an identity element. We show that a w-local mod- ule M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that w-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated am- ply Rad-supplemented left modules over any ring (not necessarily com- mutative). All rings considered in this paper will be commutative with an identity element (except for Section 5) and all modules will be left unitary modules. Unless otherwise stated R denotes an arbitrary commutative ring. Let M be an arbitrary R-module. We will denote by Rad(M) the Jacobson radical of M. A submodule L of M is called small in M (notation L ≪ M) if M 6 L+N for every proper submodule N of M. The annihilator of M in R will be denoted by AnnR(M) = {� ∈ R : �x = 0 for all x ∈ M} and for every element x of M, the annihilator of x is denoted by AnnR(x) = {� ∈ R : �x = 0}. A module M is said to be radical if Rad(M) = M. The sum of all radical submodules of a module M will be denoted by P(M). A module M is said to be reduced if P(M) = 0. We say that the ring R is reduced if the R-module RR is reduced. For submodules U and V of a module M, the submodule V is said to be a Rad- supplement of U in M if U+V = M and U∩V ⊆ Rad(V ). A module M is called Rad-supplemented if every submodule of M has a Rad-supplement in M. On the other hand, a submodule N of M is said to have ample Rad-supplements in

In a recent paper, Aydoǧdu and López-Permouth have defined a module M to be N-subinjective if every homomorphism N→M extends to some E(N)→M, where E(N) is the injective hull of N. Clearly, every module is subinjective relative to any injective module. Their work raises the following question: What is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules, namely injectives? We show, using a dual opposite injectivity condition, that such a ring R is isomorphic to the direct product of a semisimple Artinian ring and an indecomposable ring which is (i) a hereditary Artinian serial ring with J2 = 0; or (ii) a QF-ring isomorphic to a matrix ring over a local ring. Each case is viable and, conversely, (i) is sufficient for the said property, and a partial converse is proved for a ring satisfying (ii). Using the above mentioned classification, it is also shown that such rings coincide with the fully saturated rings of Trlifaj except, possibly, when von Neumann regularity is assumed. Furthermore, rings and abelian groups which satisfy these opposite injectivity conditions are characterized., TUBITAK

The aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity. © 2019 World Scientific Publishing Company.

We prove that a module M is cofinitely weak supplemented or briefly cws (i.e., every submodule N of M with M/N finitely generated, has a weak supplement) if and only if every maximal submodule has a weak supplement. If M is a cws-module then every M-generated module is a cws-module. Every module is cws if and only if the ring is semilocal. We study also modules, whose finitely generated submodules have weak supplements., TÜBİTAK

A submodule N of a right R-module M is called coneat iffor every simple right R-module S, any homomorphism N → S can beextended to a homomorphism M → S. M is called coneat-flat if thekernel of any epimorphism Y →M →0 is coneat in Y. It is proven that(1) coneat submodules of any right R-module are coclosed if and only ifR is right K-ring; (2) every right R-module is coneat-flat if and only ifR is right V-ring; (3) coneat submodules of right injective modules areexactly the modules which have no maximal submodules if and only ifR is right small ring. If R is commutative, then a module M is coneat-flat if and only if M + is m-injective. Every maximal left ideal of R isfinitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring Ris perfect if and only if every coneat-flatmodule is projective. We also study the rings over which coneat-flat andflat modules coincide. 1. IntroductionA subgroup A of an abelian group B is said to be neat in B if pA = A∩pBfor every prime integer p. The notion of neat subgroup was generalized tomodules by Renault (see, [12]). Namely, a submodule N of a right R-moduleM is called neat in M, if for every simple right R-module S, Hom(S,M) →Hom(S,M/N) → 0 is epic. Dually, in [8], a submodule N of a right R-moduleM is called coneat in M if Hom(M,S) → Hom(N,S) → 0 is epic for everysimple right R-module S. The notions of neat and coneat are coincide overthe ring of integers. By [8, Theorem], the commutative domains over whichneat and coneat submodules coincide are exactly the domains with finitely gen-erated maximal ideals (i.e., N-domains). This result was extended to certaincommutative rings in [5]. Recently, modules related to neat and coneat sub-modules are considered by several authors. In [5], a right R-module M is calledabsolutely neat (resp. coneat) if M is a neat (resp. coneat) submodule of anymodule containing it. According to [16], a right R-module M is m-injective

Let R be a ring with identity. We prove that, the flat cover of any simple right R-module is projective if and only if R is semilocal and J(R) is cotorsion if and only if R is semilocal and any indecomposable flat right R-module with unique maximal submodule is projective.

Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a no-local Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule. Зошiнгер вивчав модулi, радикали яких мають доповнення, i назвав цi модулi радикально-доповненими. Мотивуючись цим, будемо називати модуль сильно радикально доповненим (або, скорочено, srs-модулем) якщо кожен пiдмодуль, що мiстить радикал, має доповнення. Доведено, що кожен (скiнченнопороджений) лiвий модуль є srs-модулем тодi i тiльки тодi, коли кiльце є лiвим (напiв)досконалим. Над локальною дедекiндовою областю srs-модулi та радикально доповненi модулi збiгаються. Над нелокальною дедекiндовою областю srs-модуль є сумою свого пiдмодуля скруту i радикального пiдмодуля.

In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules. We prove that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that π-projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive ⊕-supplemented module is quasi-discrete. © Indian Academy of Sciences.

In a recent paper by Wang and Ding, it was stated that any ring which is generalized supplemented as a left module over itself is semiperfect. The purpose of this note is to show that Wang and Ding’s claim is not true and that the class of generalized supplemented rings lies properly between the classes of semilocal and semiperfect rings. Moreover, we propose a corrected version of the theorem by introducing a wider notion of ‘local’ for submodules.

Zhou defined $\delta$-semiperfect rings as a proper generalization of semiperfect rings. The purpose of this paper is to discuss relative notions of supplemented modules and to show that the semiperfect rings are precisely the semilocal rings which are $\delta$-supplemented. Module theoretic version of our results are obtained.

Let tau be a radical for the category of left R-modules for a ring R. If M is a tau-coatomic module, that is, if M has no nonzero tau-torsion factor module, then tau(M) is small in M. If V is a tau-supplement in M, then the intersection of V and tau(M) is tau(V). In particular, if V is a Rad-supplement in M, then the intersection of V and Rad(M) is Rad(V). A module M is tau-supplemented if and only if the factor module of M by P-tau(M) is v-supplemented where P-tau(M) is the sum of all tau-torsion submodules of M. Every left R-module is Rad-supplemented if and only if the direct sum of countably many copies of R is a Radsupplemented left R-module if and only if every reduced left R-module is supplemented if and only if RIP(R) is left perfect where P(R) is the sum of all left ideals I of R such that RadI = I. For a left duo ring R, R is a Rad-supplemented left R-module if and only if R/P(R) is semiperfect. For a Dedekind domain R, an R-module M is Bad-supplemented if and only if M/D is supplemented where D is the divisible part of M.

A necessary and sufficient condition is obtained for a right pseudo semisimple ring to be left pseudo semisimple. It is proved that a right pseudo semisimple ring is an internal exchange ring. It is also proved that a right and left pseudo semisimple ring is an SSP ring, TUBITAK (the Scientific and Technical Research Council of Turkey)

In this note we show that a ring $R$ is left perfect if and only if every left $R$-module is weakly supplemented if and only if $R$ is semilocal and the radical of the countably infinite free left $R$-module has a weak supplement.

It is shown that weakly supplemented modules need not be closed under extension (i.e. if $U$ and $M/U$ are weakly supplemented then $M$ need not be weakly supplemented). We prove that, if $U$ has a weak supplement in $M$ then $M$ is weakly supplemented. For a commutative ring $R$, we prove that $R$ is semilocal if and only if every direct product of simple $R$-modules is weakly supplemented.

International audience; We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance Dirichlet L-functions, involving in particular the study of inverse Mellin transforms); we also give a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions.

In this course, we introduce the main notions relative to the classical theory of modular forms. A complete treatise in a similar style can be found in the author’s book joint with Stromberg (Cohen and Stromberg, Modular Forms: A Classical Approach, Graduate Studies in Math. 179, American Math. Soc. (2017) [1]).