Injective Hull Research Articles

Quantum Logic in Intuitionistic Perspective

In their seminal paper Birkhoff and von Neumann revealed the following dilemma: [...] whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic. In this paper we eliminate this dilemma, providing a way for maintaining both. Via the introduction of the "missing" disjunctions in the lattice of properties of a physical system while inheriting the meet as a conjunction we obtain a complete Heyting algebra of propositions on physical properties. In particular there is a bijective correspondence between property lattices and propositional lattices equipped with a so called operational resolution, an operation that exposes the properties on the level of the propositions. If the property lattice goes equipped with an orthocomplementation, then this bijective correspondence can be refined to one with propositional lattices equipped with an operational complementation, as such establishing the claim made above. Formally one rediscovers via physical and logical considerations as such respectively a specification and a refinement of the purely mathematical result by Bruns and Lakser (1970) on injective hulls of meet-semilattices. From our representation we can derive a truly intuitionistic functional implication on property lattices, as such confronting claims made in previous writings on the matter. We also make a detailed analysis of disjunctivity vs. distributivity and finitary vs. infinitary conjunctivity, we briefly review the Bruns-Lakser construction and indicate some questions which are left open.

SCr Modules over Local Rings

We show that the following two conditions, for each integer r≥1, are equivalent for a finitely generated module M over a complete Noetherian local ring (R,m):•(a) For each integer ≥1, there is a submodule ⊂m such that / is embeddable in , where denotes the injective hull of the residue field /m.•(b) Either ⊂, or else (/m,)=< and there is no prime ideal p such that /p=1 and (/p) is embeddable in .This is an extension of a result of M. Hochster (1977, Trans. Amer. Math. Soc.231, 463–488), which is the case r=1. We show other results related to these conditions.

Purity and Equational Compactness of Projection Algebras

The notions of purity and equational compactness of universal algebras have been studied by Banaschewski and Nelson. Also, Banaschewski deals with these notions in the special case of G-sets for a group G. In this paper we study these and related concepts in the category PRO of projection algebras, that is in N∞-sets, for the monoid N∞ with the binary operation m.n=min {m,n}. We show that every monomorphism in PRO is pure and hence every equationally compact projection algebra is in fact injective. Then, we introduce the notions of s-purity and s-compactness by which we characterize the retractions and hence equationally compact projection algebras. And, among other results, we show that equationally compact, injective, and complete projection algebras are the same. Finally, we characterize (pure-)essential monomorphisms and construct the Equationally Compact Hulls, equivalently the Injective Hulls, of projection algebras. These results, among other things, generalize the main results of Guili, regarding completeness and s-injectivity in the category PROs of separated projection algebras.

Strongly rational comodules and semiperfect Hopf algebras over QF rings

Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or — equivalently — the (left) integrals form a free R-module of rank 1.

On injective hulls of simple modules

We characterize a ring over which every left module of finite length has an injective hull of finite length. Using this, we show that finite normalizing extensions of such a ring also have the same property. We also consider rings having the property that the injective hull of every simple module is artinian. We show that certain noncommutative noetherian rings have this property.

Variations of purity in abelian group theory

In a collection of classical papers, R. Nunke studied radicals R on the category of abelian groups constructed using extensions of the integers. In particular, the sequences in RExt AG were called R-pure. In this paper several types of purity related to R-purity are discussed. Projective and injective resolutions of an arbitrary group are constructed, providing interesting examples of dual classes of groups. One particular variation, that of R ∗ -purity, is shown to have many of the advantages of R-purity, while suffering from fewer of its drawbacks, e.g., every group has an R ∗ -injective hull and the class of R ∗ -projectives is closed under arbitrary subgroups. The most important example is R= p α , where α is an ordinal. When α is a limit, p α -purity and its generalizations are closely related to the completion of groups in the α-topology. Although these are the motivating examples, many of the results are stated in a substantially more general context.

Interpolation theory and measures related to operator ideals

A1Departamento de Análisis Matemaático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain E-mail: cobos@eucmax.sim.ucm.es A2Departamento de Economia Aplicada Cuantitativa, Facultad de C.C. Económicas y Empresariales, Universidad Nacional de Educación a Distancia, c/Senda del Rey s/n, 28040 Madrid, Spain E-mail: amanzano@sr.uned.es A3Departamento de Matematica Aplicada, E.T.S. Ingenieros Industriales Universidad de Vigo, 36200 Vigo, Spain E-mail: antonmar@uvigo.es

Injective Hulls in POSV

We determine the injective objects and hulls in the category POSV. This category is similar to the one of join semilattices but contains all partially ordered sets. The results of this paper have applications, for instance, in the theory of (generalized) ultrametric spaces.

CO-SEMISIMPLE MODULES AND GENERALIZED INJECTIVITY

Let $\cal F$ be a left Gabriel topology on a ring $R$ and $\cal X$ be a special class of left $R$-modules (for example, the class of all quasi-continuous left $R$-modules in $\sigma[M]$, etc.). Suppose that all left $R$-modules in $\cal X$ are $\cal F$-injective. Then, it is proved in this paper that a left $R$-module $M$ is $\cal F$-co-semisimple (that is, every $\cal F$-cocritical left $R$-module $C$ in $\sigma[M]$ is dense in its $M$-injective hull) if and only if every $\cal F$-torsionfree $\cal F$-finitely cogenerated left $R$-module $N$ in $\sigma[M]$ is dense in its some essential extensions which are in $\cal X$. As a corollary we show that a left $R$-module $M$ is co-semisimple if and only if every finitely cogenerated left $R$-module in $\sigma [M]$ is continuous (or quasi-continuous, or direct-injective, etc.)

Internal injectivity of Boolean algebras in MSet

This paper deals with the internal notion of injectivity for Boolean algebras in the topos of M-sets. Given that, for ordinary Boolean algebraas, injectivity is the same as completeness (Sikorski’s theorem) and the injective hull is the same as normal completion, we investigate here how the internal notion of completeness relates to internal injectivity. Further, we consider the internal injectivity of the initial Boolean algebra 2 which is equivalent to the prime ideal theorem for Boolean algebras in this topos. Before we turn specificially to Boolean algebras, we develop the bassic general facts concerning internal injectivity in MSet for arbitrary equational classes of algebras.

CO-SEMISIMPLE MODULES AND GENERALIZED INJECTIVITY

Let $\cal F$ be a left Gabriel topology on a ring $R$ and $\cal X$ be a special class of left $R$-modules (for example, the class of all quasi-continuous left $R$-modules in $\sigma[M]$, etc.). Suppose that all left $R$-modules in $\cal X$ are $\cal F$-injective. Then, it is proved in this paper that a left $R$-module $M$ is $\cal F$-co-semisimple (that is, every $\cal F$-cocritical left $R$-module $C$ in $\sigma[M]$ is dense in its $M$-injective hull) if and only if every $\cal F$-torsionfree $\cal F$-finitely cogenerated left $R$-module $N$ in $\sigma[M]$ is dense in its some essential extensions which are in $\cal X$. As a corollary we show that a left $R$-module $M$ is co-semisimple if and only if every finitely cogenerated left $R$-module in $\sigma [M]$ is continuous (or quasi-continuous, or direct-injective, etc.)

On uniform dimensions of ideals in right nonsingular rings

For any ( S, R)-bimodule M, one can define an invariant d( M) by taking the supremum of n for which there exists a direct sum of nonzero subbimodules N = M 1 ⊕ M 2 ⊕ … ⊕ M n such that N is essential in M as a right R-submodule. This invariant is a sort of hybrid between the right uniform dimension and the 2-sided uniform dimension. In this paper, we study the ideal structure of a right nonsingular ring R terms of the ideal structure of Q max r ( R) by working with the invariant d( I) = d( R I R ) for ideals I ⊂ R. The family F( R) of ideals I for which there exists an ideal J ⊂ R with I ⊕ J ⊂ e R r is characterized in various ways, and for I ∈ F( R), the invariant d( I) is related to the direct product decomposition of the ring E( I R ) (injective hull) in Q max r ( R). It is shown that d( I) is very well-behaved for the ideals I ∈ F( R) and various results are obtained on the relationship between d( I), u. dim( R I R ) and u. dim( I R ).

Rings whose Injective Hulls are Cogenerators

Rings whose Injective Hulls are Cogenerators

Test ideals in quotients of $F$-finite regular local rings

Let S be an F -finite regular local ring and I an ideal contained in S. Let R = S/I. Fedder proved that R is F -pure if and only if (I [p] : I) * m[p]. We have noted a new proof for his criterion, along with showing that (I [q] : I) ⊆ (τ [q] : τ), where τ is the pullback of the test ideal for R. Combining the the F -purity criterion and the above result we see that if R = S/I is F pure then R/τ is also F -pure. In fact, we can form a filtration of R, I ⊆ τ = τ0 ⊆ τ1 ⊆ . . . ⊆ τi ⊆ . . . that stabilizes such that each R/τi is F -pure and its test ideal is τi+1. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let R = T/I, where T is either a polynomial or a power series ring and I = P1 ∩ . . . ∩ Pn is generated by monomials and the R/Pi are regular. Set J = Σ(P1 ∩ . . . ∩ Pi ∩ . . . ∩ Pn). Then J = τ = τpar. This paper concerns the study of the test ideal in F -finite quotients of regular local rings. Test elements play a key role in tight closure theory. Once known, they make computing tight closures of ideals and modules easier. In fact, in an excellent Gorenstein local ring with an isolated singularity, R/τ ∼= Hom(I∗/I, E) where I is generated by a system of parameters that are test elements and E is the injective hull of R (see [Hu1] and [S1]). We also know for parameter ideals I that I : τ = I∗. Thus knowing τ is basically equivalent to knowing the tight closure of a system of parameters which is contained in the test ideal. A recent paper of Huneke and Smith [HS] links tight closure to Kodaira vanishing for graded rings R with characteristic either 0 or p where p 0. Recall that the a-invariant for a graded ring R, denoted a, is equal to −min{i|[ωR]i 6= 0}, where ωR is the canonical module for R. If R is Gorenstein, then ωR = R(a). Huneke and Smith prove that the test ideal is exactly the ideal generated by elements of degree greater than the a-invariant of R if and only if a strong Kodaira vanishing holds on R. A recent paper of Hara [Ha] confirms that this strong Kodaira vanishing holds in finitely generated algebras over a field of characteristic zero. In this paper we study test ideals of F -finite rings which are reduced quotients of a regular local ring. Reduced quotients of F -finite regular local rings have been studied by both Fedder [Fe] and Glassbrenner [Gl]. Fedder’s work concerns F purity aspects of these rings, and Glassbrenner’s results use Fedder’s techniques to examine strong F -regularity. The object that plays a key role in their work is Received by the editors November 4, 1996. 1991 Mathematics Subject Classification. Primary 13A35.

ALGEBRAIC HULLS OF ADJOINT FUNCTORS AS INJECTIVE HULLS

For each adjoint functor U: A → X where X is an (ϵ, M)-category having enough ϵ-projectives, we construct an (ϵ, M)-algebraic hull E: (A, U) → (Â, Û), i.e., (Â, Û) is (epsiv; M)-algebraic and E has a certain denseness property. We show that there is a conglomerate of functors over X with respect to which the (ϵ M)-algebraic categories are exactly the injective objects and characterize (ϵ M)-algebraic hulls as injective hulls.

The Residue Complex of a Noncommutative Graded Algebra

Ž . where X rX : X is the set of points of dimension q the q-skeleton q qy1 Ž . Ž . and K x is an injective hull of the residue field k x . The coboundary A Ž . Ž . operator d : K x a K y is nonzero precisely when y is an immediate A A w x specialization of x. For a discussion of the commutative theory see RD w x and Ye2 . In this paper we propose a definition of the residue complex R ? of a noncommutative Noetherian graded k-algebra A s k [ A [ A [ ??? . 1 2 Ž . We begin, in Section 1, with the generalized Auslander]Gorenstein A-G condition. This condition can be checked whenever A has a dualizing Ž . complex; if A is Gorenstein i.e., has finite injective dimension it reduces

On Quasi-Harada Rings

M. Harada (“Ring Theory, Proceedings of 1978 Antwerp Conference,” pp. 669–690, Dekker, New York, 1979) studied the following two conditions:(*) Every non-small leftR-module contains a non-zero injective submodule.(*)* Every non-cosmall rightR-module contains a non-zero projective direct summand.K. Oshiro (Hokkaido Math. J.13, 1984, 310–338) further studied the above conditions, and called a left artinian ring with (*) a left Harada ring (abbreviated left H-ring) and a ring satisfying the ascending chain condition for right annihilator ideals with (*)* a right co-Harada ring (abbreviated right co-H-ring). K. Oshiro (Math. J. Okayama Univ.31, 1989, 161–178) showed that left H-rings and right co-H-rings are the same rings. Here we are particularly interested in the following characterization of a left H-ring given in Harada's paper above: A ringRis a left H-ring if and only ifRis a perfect ring and for any left non-small primitive idempotenteofRthere exists a non-negative integertesuch that(a)RRe/Sk(RRe) is injective for anyk∈{0,…,te} and(b)RRe/Ste+1(RRe) is a small module, whereSk(RRe) denotes thek-th socle of the leftR-moduleRe.This characterization implies(+)Ste+1(RRe) is a uniserial leftR-module for any left non-small primitive idempotentein a left H-ringR.In this paper, we generalize left H-rings by removing (+). Concretely, since a left H-ringRis also characterized by the statement thatRis left artinian and for any primitive idempotentgofRthere exist a primitive idempotentegofRand a non-negative integerkgsuch that the injective hull of the leftR-moduleRg/Jgis isomorphic toRe/Skg(RReg), whereJis the Jacobson radical ofRR, we define a more general class of rings by the condition that for any primitive idempotentgofRthere exists a primitive idempotenteofRsuch that the injective hull of the leftR-moduleRg/Jgis isomorphic toRe/{x∈Re|gRx=0}, and call it a left quasi-Harada ring (abbreviated left QH ring). In Section 1 we characterize a left QH ring by generalizing the characterizations of a left H-ring given by M. Harada and K. Oshiro. We also consider the weaker rings, left QF-2 rings and right QF-2* rings. K. Oshiro (Math. J. Okayama Univ.32, 1990, 111–118) described the connection between left H-rings and QF rings. In Section 2 we describe the connection between two-sided QH rings and QF rings.

Mutual Infective Hulls

AbstractThe mutual injective hull of an arbitrary family of modules is constructed. Applications to the calculations of quasi-injective and π-injective hulls of direct sums are given.

Relative QF-3' modules and QF-3 modules

As a generalization of QF rings, the notion of rings was introduced. Many authors have studied connections between rings, Morita duality and quotient rings. Basic part of these works can be found in [21]. Several results for rings are extended to more general classes of rings. Rings R whose injective hull E(RR) is torsionless (right QF-3' rings) and rings R in which every finitely generated submodule of the injective hull E(RR) is torsionless (according to (91, we call such rings right QF-3 rings) are such kind of rings. These rings are closely related to the dual functors, the Lambek torsion theories, and Morita duality for Grothendieck categories (see [5], [6], [7], [9] and [ID]). On the other hand, these rings are generalized to modules, as QF-3' and QF-3 modules. QF-3' modules were mainly investigated from torsion theoretical aspects in [2] and [13]. Recently QF3 modules were introduced and studied by Ohtake in [16]. He investigated a relationship between QF-3 modules and localized Morita dualities. In this paper we shall introduce and investigate relative QF-3' modules and modules. Relative QF-3' modules are generalizations of QF-3'

Maximal (semi-)lattices of fractions and injective hulls

We show that the maximal (semi-)lattice of fractions of a (semi-)lattice may be obtained as a canonical subsemilattice of the bicommutator of an injective hull of the (semi-)lattice under consideration, whithin a suitably defined category of semilattices. The construction is roughly similar to that of a maximal semigroup of quotients, but there are significant differences due to the commutativity and idempotency of (semi-)lattice operations. In particular, the construction is effective in the sense that it avoids the use of Zorn's lemma to obtain the required injective hulls.