A Characterization of Group Rings

In this paper we prove that every coseparable involutory Hopf algebra over the ring of integers Z which is a free Z-module is the group ring of some group. This result was proved independently for Hopf algebras which are finitely generated Z-modules by H.-J. Schneider [6], using similar techniques. We then give some examples of coseparable Hopf algebras over number rings which are not group algebras, and give an example of a cocommutative coseparable coalgebra over a number ring which cannot be given a multiplicative structure making it into a Hopf algebra. The Hopf algebra structure theory required for this paper is found in [1], [4], and [5]. For completeness we give proofs here of the coalgebra analogues to some “well-known” facts about separable algebras.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

On the Factoriality of Local Rings of Small Embedding Codimension

Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A has embedding codimension ≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial. This tells us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.     

EPIREFLECTIVE SUBCATEGORIES AND SEMICLOSURE OPERATORS

ABSTRACT

Let C be a category of topological spaces and continuous functions which is full, hereditary and closed under homeomorphisms and products. If A is a subclass of C, let E(A) be the full subcategory of C whose objects are the subspaces in A. In this paper we characterize the epireflective subcategories of C containing A and contained in E(A) by introducing a “semiclosure” operator which is a generalization for the “idempotent semi-limit” operator introduced by S.S. Hong (see [5]) with respect to Top _o. In case A is extensive in C, so that E(A) = C, all the extensive subcategories of C containing A are thus characterized.     

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT

In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P ε P} and H(p) = {A} if B ? A with B ε N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.     

Galois Objects of Finitely Generated Projective Hopf Algebras

Let C be the category of cocommutative coalgebras over a commutative ring R and let H be a group object in C, i.e., let H be a cocommutative Hopf algebra. Assume that H is a finitely generated, projective R-module and that the integrals (of [4]) in H* ≡ Hom_R(H, R) are cocommutative elements. We will show that any Galois H-object (as defined in [3, Def. 1.2, p. 8]) is a finitely generated, projective R-module.     

VARIETAL HULLS OF FUNCTORS

Abstract

The well known characterizations of equational classes of algebras with not necessaryly finitary operations by FELSCHER [6.7] and of categories of A-algebras for algebraic theories A in the sense of LINTON [10], esp., by means of their forgetful functors are the foundations of a concept of varietal functors U:K → L over arbitrary basecategories L. They prove to be monadic functors which satisfy an additional HOM-condition [17]. (In the case L = Set this condition is always fulfilled, see LINTON [11].)

Contrary to monadic functors, varietal functors are closed under composition. Pleasent algebraic properties of the base-category L can be ‘lifted’ along varietal functors, such as e.g. factorization properties, (co-) completeness, classical isomorphism theorems, etc.

By means of the well known EILENBERG-MOORE-algebras there is a universal monadic functor U^T:L ^T → L for any functor U: K → L, having a left adjoint F (T: = UF). But, in general, UT is not varietal. Under some suitable conditions, however it is possible, to construct a canonical varietal functor ?:R → L, the varietal hull of U. This hull has much more interesting (algebraic) properties than the EILENBERG-MOORE construction. Moreover, results of BANASCHEWSKI-HERRLICH [2] are extended.     

On the Homological Dimension of Algebras of Differential Operators

Let A be a commutative algebra over a field k, and V_A be the k-subalgebra of End_k(A) generated by End_A(A) = A and all k-derivations of A. A study of the homological properties of V_A was initiated by Hochschild, Kostant, and Rosenberg in [5], and continued by Rinehart [8], [9], Roos [11], Björk [1], Rinehart and Rosenberg [10], and others. It was proved in [5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of V_A is between r and 2r. Moreover, if k has positive characteristic, then gl.dim V_A = 2r [8]. By a recent celebrated theorem of Roos [11], gl.dim V_A = r if k has characteristic zero and A = k[x₁, …, x_r]; in this case V_A is the so-called “Weyl algebra on 2r variables”.     

Triple Cohomology and the Galois Cohomology of Profinite Groups

Given a cotriple 𝔾 = (G, ε, δ) on a category X and a functor E:X ^Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hⁿ(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hⁿ(X, Y)_𝔾 instead of Hⁿ(X, E)_G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H?ⁿ(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.     

Ascent of properties for absolutely flat homomorphisms

We show that if A – B is an absolutely flat homomorphism of consultative rings and A is arithmetical, then B is such, thus generalising a previous result (see [9]) on valutation rings.

To prove the above theorem we show first that it is true when A is a local ring and B is a local ind-étale homomorphism (in par ticular if B is a strict henselization of A ) , and we apply the following general fact:a local property which ascends to strict henselization and descends by faithful flatness ascends also by ab solutely flat homomorphlsms. This last result also applies to other properties, such as locally noetherian, geometrically unibranche, R_n,S_n, Cohen-Macaulay.     

ANOTHER CHARACTERIZATION OF BICOREFLECTIVE SUBCATEGORIES

Abstract

In this paper, the relation between the notion of a discrete functor (see [4]) and the notion of a fine functor (see [1]) is examined. As a generalization of the notion of a F-fine object (see [1]), discrete functors T: A → X are used to define K-fine objects, where K is a class of A-objects. It is shown that if T is in addition semi-topological, then (as for F-fine objects in a topological category, see [1]) the class of K-fine objects determines a bicoreflective subcategory of A. Moreover, it is shown that in co-complete, co-(well-powered) categories, the existence of bicoreflective subcategories is equivalent to the existence of functors that are both discrete and semi-topological.     

LIGHT FACTORIZATION STRUCTURES

Abstract

In this paper we investigate, for connection subcategories A of a topological category K, the concepts of A-monotone quotients and A-light sources, and characterize (1) those A, which give rise to an (A-monotone quotient, A-light)- factorization structure on K, (2) those factorization structures (C,D) on K, which are light, i.e. of the form (A-monotone quotient, A-light) for suitable A. It turns out that light factorization structures are rather rare in Top, but abundant and well-behaved in categories with hereditary quotients.     

EPIREFLECTIONS VERSUS BIREFLECTIONS FOR TOPOLOGICAL FUNCTORS

Abstract

In this paper it is proved that if T: A → X is a topological functor satisfying certain conditions, then there is a Galois Connection between the class of bireflective subcategories of A and the class of epireflective subcategories of A that are not bireflective and that are contained in the subcategory of separated objects of A. In general such a correspondence is not bijective.     

Functional Identities and Rings of Quotients

The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q _{m l} (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q _{m s} (R), but not for the symmetric Martindale ring of quotients Q _s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q _s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations.     

On Going Down For Simple Overrings II

This paper deals with two themes involving the going down (GD) behavior of the overrings of a domain R. First, as illustrated by several results in [3] and [4], hypotheses about GD often replace stronger assumptions about flatness or one-dimensionality in sets of conditions implying that R is Prüfer. Second, as shown by the two-dimensional example in [4]. Remarks (iii)], some sort of finiteness condition is required in conjunction with GD hypotheses in order for R to be Prüfer.     

DISCRETE FUNCTORS

Abstract

The concept of a T-discrete object is a generalization of the notion of discrete spaces in concrete categories. In this paper. T-discrete objects are used to define discrete functors. Characterizations of discrete functors are given and their relation to other important functors are studied. A faithful functor T: A → X is discrete iff the full subcategory B of A consisting of all T-discrete objects is (X-iso)-coreflective in A. It follows that the existence of bicoreflective subcategories is equivalent to the existence of suitable discrete functors. Finally, necessary and sufficient conditions are found such that for a given functor T: A → X, the full subcategory B of A consisting of all T-discrete A-objects is monocoreflective in A.     

Syntactic Rings

If the state set and the input set of an automaton are Ω-groups then near-rings are useful in the study of automata (see [5]). These near-rings, called syntactic near-rings, consist of mappings from the state set Q of the automaton into itself. If, as is often the case, Q bears the structure of a module, then the zerosymmetric part N₀(A) of syntactic near-rings is a commutative ring with identity. If N₀(A) is a syntactic ring then its ideals are useful for determining reachability in automata (see [1] or [2]). In this paper we investigate syntactic rings.     

Reversibility of Rings with Respect to the Jacobson Radical

Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reversible if for any a, \(b \in R\), \(ab = 0\) implies \(ba \in J(R)\). In this paper, we give some properties of J-reversible rings. We prove that some results of reversible rings can be extended to J-reversible rings for this general setting. We show that J-quasipolar rings, local rings, semicommutative rings, central reversible rings and weakly reversible rings are J-reversible. As an application it is shown that every J-clean ring is directly finite.     

Flat modules and rings finitely generated as modules over their center

A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:

1. A is a right or left distributive semiprime ring;
2. for any maximal idealM of a subringR central inA, the ring of quotientsA _M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
3. all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.