Baer Criterion Research Articles

Characterizing Topologically Dense Injective Acts and Their Monoid Connections

In this paper, we explore the concept of topologically dense injectivity of monoid acts. It is shown that topologically dense injective acts constitute a class strictly larger than the class of ordinary injective ones. We determine a number of acts satisfying topologically dense injectivity. Specifically, any strongly divisible as well as strongly torsion free S-act over a monoid S is topologically dense injective if and only if S is a left reversible monoid. Furthermore, we establish a counterpart of the Skornjakov criterion and also identify a class of acts satisfying the Baer criterion for topologically dense injectivity. Lastly, some homological classifications for monoids by means of this type of injectivity of monoid acts are also provided.

Test sets for factorization properties of modules

Baer's Criterion of injectivity implies that injectivity of a module is a factorization property with respect to\ a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring R and on additional set-theoretic hypotheses. For R commutative noetherian of Krull dimension 0 < d < \infty , we show that the assertion 'projectivity is a factorization property with respect to a single epimorphism' is independent of ZFC + GCH. We also show that if R is any ring and there exists a strongly compact cardinal \kappa > |R| , then the category of all projective modules is \kappa -accessible.

Down closed injectivity and essentialness

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for [Formula: see text]-posets, posets with an action of a pomonoid [Formula: see text] on them. They gave a criterion for dc-injectivity and studied such injectivity for [Formula: see text] itself, and for its poideals. In this paper, we give results about dc-injectivity of [Formula: see text]-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of [Formula: see text]-injectivity for [Formula: see text]-posets, where [Formula: see text] and [Formula: see text] is externally adjoined to the posemigroup [Formula: see text], we find some classes of pomonoids such that for [Formula: see text]-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of [Formula: see text]-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of [Formula: see text]-posets for some classes of pomonoids [Formula: see text].

Strongly Fully Stable Acts Relative to an Ideal

The purpose of this paper is to introduce and investigate strongly fully stable acts relative to an ideal as a concept generalizing strongly fully stable modules relative to an ideal, but is stronger than that of fully stable acts. In this study, we consider some properties and characterizations of the class of strongly fully stable acts relative to an ideal, as well as the relations between this class and other classes. Among these classes are quasi-injective acts, strongly quasi-injective acts, acts which satisfy Baer's criterion, acts satisfying the strongly Baer's criterion, and duo acts. The product of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. The coproduct of any family of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. Also, we have that strongly fully stable acts relative to an ideal are equivalent to an $S$-act that satisfies the strongly Baer's criterion relative to an ideal $I$ for cyclic subacts. The strongly fully stable act relative to $I$ is equivalent to the strongly quasi-injective act relative to the ideal $I$ and duo act with a commutative monoid.

Faith’s problem on $R$-projectivity is undecidable

In \cite{F}, Faith asked for what rings $R$ does the Dual Baer Criterion hold in Mod-$R$, that is, when does $R$-projectivity imply projectivity for all right $R$-modules? Such rings $R$ were called right testing. Sandomierski proved that if $R$ is right perfect, then $R$ is right testing. Puninski et al.\ \cite{AIPY} have recently shown for a number of non-right perfect rings that they are not right testing, and noticed that \cite{T2} proved consistency with ZFC of the statement {\lq}each right testing ring is right perfect{\rq} (the proof used Shelah's uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith's question above is undecidable in ZFC. We also provide examples of non-right perfect rings such that the Dual Baer Criterion holds for {\lq}small{\rq} modules (where {\lq}small{\rq} means countably generated, or $\leq 2^{\aleph_0}$-presented of projective dimension $\leq 1$).

On unitary down-closed regular monomorphisms of pomonoid actions

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.

Sequentially pure injectivity

Injectivity with respect to pure monomorphisms was studied before in categories of modules and categories of acts. In this paper we study this notion with respect to sequentially pure monomorphisms.Although the Baer criterion for injectivity (weakly injectivity implies injectivity) is true for modules over a ring (with an identity), it is an open problem for acts over a semigroup S (with or without identity). In fact, we are not aware of any type of weak injectivity implying injectivity of S-acts, in general, other than the Skornjakov-Baer criterion, which says that injectivity with respect to subacts of cyclic acts implies injectivity with respect to all monomorphisms.M.M. Ebrahimi and M. Mahmoudi in some papers have shown this criterion to be true for some special S.

Some remarks on Hom-modules and Hom-path algebras

This paper deals with injective and projective right Hom-H-modules for a Hom-algebra H. In particular, Baer Criterion of injective Hom-module is obtained, and it is shown that HomModH is an Abelian category. Next, the authors define Hom-path algebras and construct Hom-path algebras of some quivers.

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.

Weakly Injective BCK-Modules

We introduce the notion of weakly injective BCK-module and show that Baer's criterion holds for weakly injective BCK-modules but not for injective BCK-modules in general. We also provide examples and counterexamples of weakly injective BCK-modules.

On the applicability to semirings of two theorems from the theory of rings and modules

Problems concerning the extension of the Baer criterion for injectivity and embedding theorem of an arbitrary module over a ring into an injective module to the case of semirings are treated. It is proved that a semiring S satisfies the Baer criterion and every S-semimodule can be embedded in an injective semimodule if and only if S is a ring.

On the Baer Criterion for Acts Over Semigroups

It is known that although the Baer Criterion for injectivity holds for modules over rings with unit, it is not true for acts over an arbitrary monoid. Seeking a characterization for the Baer Criterion to hold for acts over a monoid, in this article, using the notion of completeness introduced by Giuli, we find some classes of monoids such that for acts over them the Baer Criterion holds.

Injective Hulls of Acts over Left Zero Semigroups

In this paper, using the notion of “Cauchy sequences”, we study the injectivity of acts over left zero semigroups, and give some equivalent conditions to it. We see that Baer criterion is true if an identity is adjoined to the semigroup. Further, we find injective hulls of acts over left zero semigroups. We hope that the technique used here will be useful to study injectivity of acts over some other more general semigroups.

Baer Criterion for Injectivity of Projection Algebras

Projection algebras are M-sets for the monoid M = (ℕ∞, min) which are used by computer scientists for algebraic specification of process algebras. In contrast to the case of modules it is well known that the Baer Criterion does not generally hold for injectivity of M-sets for an arbitrary monoid M. Here, introducing a closure operator, we prove that the Baer Criterion does hold for injectivity of projection algebras.

EXPLICIT DESCRIPTIONS OF THE INDECOMPOSABLE INJECTIVE MODULES OVER JATEGAONKAR'S RINGS

ABSTRACT In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Injectivity and injective hulls of Abelian groups in a localic topos

We prove the analogue of the Baer Criterion for injectivity in the category AbShℒ of abelian groups in a topos of sheaves on a locale, that is, we show A is injective in AbShℒ if and only if it is injective relative to all S ↣ Zℒ where Zℒ is the group of integers in Shℒ. for a well-ordered locale we describe the injective hulls in AbShℒ in terms of injective hulls in Ab. Further we show that the global functor A → AE preserves injective hulls if and only if ℒ is a finite boolean locale. Finally We characterise injectives in AbShℒ for some special locales.