Simple Socle Research Articles

QF-1 algebras with faithful direct sums of uniserial modules

Let M be a left unital module over a ring R. The module M is called balanced if the canonical ring homomorphism from R into the double centralizer of M is surjective. Following Thrall [lOI R is said to be QF-1 if every finitely generated faithful R-module is balanced. In [6] Ringel proved that if an algebra with square zero radical is QF-1, then it is a Tachikawa algebra (= an algebra of local-colocal type), i.e., an algebra each of whose indecomposable modules has either the simple top or the simple socle. Also, Tachikawa [9] proved that if an algebra whose left regular module is serial (= a left serial algebra) is QF-1, then it is a Tachikawa algebra (particularly, an algebra of left colocal type). Here, a serial module is a direct sum of uniserial modules (a module is uniserial in case it has the unique composition series). It should be noted that an algebra with square zero radical, as well as a left serial algebra, has a faithful serial module. The purpose of this paper is to give a generalization of both of the above results by Ringel and by Tachikawa in the case where the ground field of an algebra is algebraically closed. Indeed we shall prove

Structure of cohomology of line bundles on [formula omitted] for semisimple groups

Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field k of characteristic p > 0, T a maximal torus of G and B a Borel subgroup containing T . Each weight in X(T ) determines a line bundle on the flag varietyG/B. It turns out that cohomology of the line bundle is isomorphic to the derived functor of the induction functors from the category of B-modules to the category of G-modules for each 1-dimensional B-module defined by a weight in X(T ). Further, IndGBλ = H (G/B, λ) turns out to be the dual of a Weyl module. One of the main problems is to calculate the characters of the irreducible G-modules. For p = 0, the character of the irreducible G-module of highest weight λ is given by Weyl’s character formula and the G-module structures of the cohomology of line bundles are well understood. However, for p > 0, the story is quite different and many of the results remain conjectural. Since characters of Weyl modules are given by Weyl’s character formula, understanding the structure ofH(G/B, λ) turns out to be the main problem. It is also interesting to understand the structure of the higher cohomology, which might help us to understand H. H. Andersen has a series of papers toward the understanding of the higher cohomology, such as the simple socle of H and filtrations of H i [4]. In [5] , using the representations of infinitesimal subgroup schemes of G, he proved generically that H(G/B,w ·λ) has simple socle and simple head and their highest weights can be calculated. In [6], he proved that the socle series of H(λ) comes generically from the

Linear Groups and Distance-transitive Graphs

A detailed treatment is given of the graphs on which a group with simple socle PSL ( n , q ) acts primitively and distance-transitively.

On artinian $QF-3$,$1$-Gorenstein rings

A noetherian ring is called 1-Gorenstein if it has the self-injective dimension at most one on both sides. A well known example of artinian QF-3, 1-Gorenstein rings is the triangular matrix ring over a QF ring, which is QF-2, that is, every indecomposable projective module has the simple socle. Conversely Sumioka [11] characterized such a ring as an artinian QF-3, 1-Gorenstein ring with QF maximal quotient ring. But an artinian QF-3, 1Gorenstein ring has not necessarily the QF maximal quotient ring (see §4).On the other hand, Sumioka's result is a generalization of Harada's characterization of artinian QF-3 hereditary rings, which states that a connected artinian ring is QF-3 hereditary if and only if it is Morita equivalent to a triangular matrix ring over a division ring (cf. [3]). Our results in the present paper are closely related to their results mentioned above. First we shall deal with artinian QF-3 hereditary rings, which were investigated by Harada [3] and Iwanaga [41. Our result is as follows.

The Structure of QF-3 Rings

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.