On semirings satisfying the Baer criterion

It is well-known that for modules over rings the Baer injectivity criterion takes place. In this paper we prove that under one additional condition this criterion is also valid for modules over semirings. We prove that a semiring S satisfies the Baer criterion if and only if all injective (with respect to one-sided ideals of S) semimodules satisfy the above condition. We propose a newmethod for constructing semirings satisfying the Baer criterion.     

On unitary down-closed regular monomorphisms of pomonoid actions

Abstract

In this paper we characterize injective objects in the category of S-posets and S-poset maps for a pomonoid S, with respect to the class of unitary down-closed embeddings. Also, the behaviour of this notion of injectivity with respect to products and coproducts is studied. Then we introduce the notion of weakly regular d-injectivity in arbitrary slices of the category of S-posets, which is applied to investigate the Baer criterion. Finally we present an example to show that these objects are not regular injective, in general.     

Quasi-injective dimension

Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.     

Selforthogonal Modules with Finite Injective Dimension III

Let R be a left Noetherian ring, S a right Noetherian ring and _R U a generalized tilting module with S?=?End( _R U). We give some equivalent conditions that the injective dimension of U _S is finite implies that of _R U is also finite. As an application, under the assumption that the injective dimensions of _R U and U _S are finite, we construct a hereditary and complete cotorsion theory by some subcategories associated with _R U.     

Supplement Submodules of Injective Modules

An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R _R is F-almost injective.     

Direct injective modules

Throughout this paperR will denote a ring with idenity element andM a unitary right module overR. AnR-moduleM is said to be direct injective if and only if given direct summandN ofM with injectioni _N:N→M and a monomorphismg:N→M, there exists an endomorphismf ofR-moduleM such thatfg=i _N. In this paper we investigate properties of direct injective modules, and obtain the following results on direct injective modules.

1. We establish the necessary and sufficient condition for a module to be direct injective.
2. We show that the answer on problem of Krull-Schmidt-Matlis is in the affirmative in caseR-moduleM is extending direct injective.
3. We prove that extending direct injectivity of module implies same properties of its direct summands.

     

Regular semiartinian rings

We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

Modules with Few Types Over A Serial Ring and Over A Commutative Prüfer Ring

It is proved that the theory of a module M over a countable serial ring has few models iff M is ∑-pure—injective iff the theory of M has few types. It is also proved that the theory of a module M over a countable commutative Prüfer ring has few types if and only if M is ∑-pure—injective.     

On Maximal Injectivity

A right R-module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R-homomorphism f ： m → E can be extended to an R-homomorphism f^1 ： R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R-module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith＇s conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith＇s conjecture, which asserts that every injective right R-module over any left perfect right self-injective ring R is the injective hull of a projective submodule.     

Gorenstein injective and strongly cotorsion modules

By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of _R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).     

Equivariant completeness and regular injectivity of S-posets

Abstract

In this paper, considering the actions of a pomonoid S on posets, namely S-posets, we study some relations between equivariant completeness and regular injectivity of S-posets which lead to some homological classification results for pomonoids. In particular, we show that regular injectivity implies equivariant completeness, but the converse is true only if S is left simple. Finally, it is proved that regularly injective S-posets are exactly the complete and cofree-retract ones. Among other results, we also see that the Skornjakov and Baer criteria fail for regular injectivity of S-posets.     

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.     

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.     

Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

S-内射模及S-内射包络

设R是环.设S是一个左R-模簇,E是左R-模.若对任何N∈S,有Ext_R~1(N,E)=0,则E称为S-内射模.本文证明了若S是Baer模簇,则关于S-内射模的Baer准则成立;若S是完备模簇,则每个模有S-内射包络;若对任何单模N,Ext_R~1(N,E)=0,则E称为极大性内射模;若R是交换环,且对任何挠模N,Ext_R~1(N,E)=0,则E称为正则性内射模.作为应用,证明了每个模有极大性内射包络.也证明了交换环R是SM环当且仅当T/R的正则性内射包e(T/R)是∑-正则性内射模,其中T=T(R)表示R的完全分式环,当且仅当每一GV-无挠的正则性内射模是∑-正则性内射模.     

Axiomatizability of the class of weakly injective <Emphasis Type="Italic">S</Emphasis>-acts

The concepts of weakly injective, fg-weakly injective, and p-weakly injective S-acts generalize that of injective S-act. We study the monoids S over which the classes of weakly injective, fg-weakly injective, and p-weakly injective S-acts are axiomatizable. We prove that the class of p-weakly injective S-acts over a regular monoid is axiomatizable.