Finitistic n-Self-Cotilting Modules

A right R-module M is said to be finendo provided that as a left module over its endomorphism ring it is finitely generated. Let B = End MR. Then Bn → M → 0 is exact for some finite integer n. This implies (and when M is quasi-injective is implied by) the exactitude of 0 → R/annRM → Mn. This implies that a finendo module is quasi iff injective modulo annihilator (1). Similarly for any finitely generated module over a commutative ring. Any finendo quasi-injective module is injective over its biendomorphism ring (17B). A ring R is right Artinian iff every quasi-injective right module is finendo (17). This supplies the converse of a theorem of K. Fuller [69]. Moreover, a ring R is right self-injective iff every finendo faithful (injective) right R-module generates mod-R (18B). Azumaya defined a right PF ring as one such that every faithful right R-module generates mod-R. We characterize a right PF ring as a right self-injective ring such that every faithful (quasi) injective right R-module is finendo (38). Then, if every factor ring is PF, the ring must be right Artinian, hence uniserial (40). In general, a factor ring R/A of a right PF ring is right PF iff the left annihilator of A is Rz = zR for some z e R (39).

In a category of modules the notions of />injectivity (with respect to a torsion radical p) and quasi-injectivity can be generalized to a notion of injectivity with respect to two preradicals simultaneously. Using this general definition an analog of Baer's condition for injectivity is obtained, as well as other generalizations of results for injective and quasi-injective modules. An alternate approach (not requiring the existence of injective envelopes) is given for abelian categories, with the results stated in dual form for projectivity.

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

Introduction. In '[12] Thrall introdcuced three generalizations of the quasiFrobenius (=QF) algebras of Nakayama [9], [10]. In this paper we shall be concerned with ring theoretic generalizations of two of these Thrall algebras-namely QF-2 algebras and QF-3 algebras. If R is a ring then a one-sided ideal of R is primitive in case it is generated by a primitive idempotent, and an R-module is minimalfaithful in case it is faithful and has no proper faithful direct summand. Extending Thrall's original definitions to (two-sided) artinian rings we have: QF-2 rings: An artinian ring is QF-2 in case each of its primitive one-sided ideals has a simple socle. QF-3 rings: An artinian ring is QF-3 in case it has (to within isomorphism) a unique minimal faithful module. It is not difficult to show that QF rings are both QF-2 and QF-3 (see [2, ??58-59]). Moreover, Thrall [12] has shown that QF-2 algebras are QF-3 but not necessarily QF. Most of the information about QF-2 and QF-3 rings is limited to finite dimensional algebras (see [8], [12], [13]). Two notable exceptions generalize to QF-3 rings results known to hold for QF-3 algebras almost from their inception. Specifically, Jans [7] has characterized QF-3 rings as those artinian rings whose injective hulls are projective and Harada [5] has shown that the QF-3 property is actually two-sided (i.e., a QF-3 ring has a unique minimal faithful right module). In ?2 of this paper we obtain ideal theoretic characterizations of the injective projective modules (and hence of the unique minimal faithful module) over left QF-3 rings. Our main results appear in ?3. With the aid of Morita's duality theorems [8] we obtain characterizations of QF-3 rings that are analogous to Nakayama's original definition of QF rings in terms of socles of primitive one-sided ideals [10], his characterization of QF-rings in terms of the double annihilator property for one-sided ideals [10], and the fact (see [8, ?14]) that QF rings are precisely those artinian rings for which the functor HomR ( , R) provides a duality between the categories of finitely generated and finitely generated right R-modules.

Duality theory for convergence groups

Nazarov [Naz96] introduced an infinite dimensional algebra, which he called theaffine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rankrn(2n−1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the$\mathfrak{p}$-local and$\mathfrak{p}$-torsion subcategories of the derived category, for each homogeneous prime ideal$\mathfrak{p}$arising from the action of a commutative ring via Hochschild cohomology.

The general problem of linear approximation is concerned with the characterization of the range of a given functional on the subset of a linear space which is defined by the data. In most applications that domain is the intersection of a sphere and a linear variety so that the conventional techniques based upon classical inequalities that are sharp on a sphere cannot yield optimal appraisals. In Hilbert spaces the required characterization is given by the well known hypercircle inequality which is based uponPythagoras' theorem and therefore cannot be extended to the general case where the norm is not expressible in terms of an inner product. The optimization theory in seminormed spaces and a practical method of solution are developed by resorting to duality theorems and more specially to the theory of convex bodies. A simple but nontrivial application is given by way of illustration.

The varieties of semi-conformal vectors of affine vertex operator algebras

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.