Radical Square Research Articles

The singularity category of an algebra with radical square zero

To an artin algebra with radical square zero, a regular algebra in the sense of von Neumann and a family of invertible bimodules over the regular algebra are associated. These data describe completely, as a triangulated category, the singularity category of the artin algebra. A criterion on the Hom-finiteness of the singularity category is given in terms of the valued quiver of the artin algebra.

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.

The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero

We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers.

When are torsionless modules projective?

In this paper, we study the problem when a finitely generated torsionless module is projective. Let Λ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless Λ-module M is projective if Ext Λ 1 ( M , M ) = 0 . For a commutative Artinian ring Λ, a finitely generated torsionless Λ-module M is projective if the following conditions are satisfied: (1) Ext Λ i ( M , Λ ) = 0 for i = 1 , 2 , 3 ; and (2) Ext Λ i ( M , M ) = 0 for i = 1 , 2 . As a consequence of this result, we have that for a commutative Artinian ring Λ, a finitely generated Gorenstein projective Λ-module is projective if and only if it is selforthogonal.

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group $\mathcal{G}$ is largely determined by a linearly reductive subgroup scheme $\hat{\mathcal{G}}$ of SL(2), with the McKay quiver of $\hat{\mathcal{G}}$ relative to its standard module being the Gabriel quiver of the principal block $\mathcal{B}_0(\mathcal{G})$ . The graphs underlying these quivers are extended Dynkin diagrams of type $\tilde{A}, \tilde{D}$ or $\tilde{E}$ , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

Auslander–Reiten Orbit Algebras for Self-Injective Nakayama Algebras

The main result of this paper is the following: Let A be a self-injective Nakayama K-algebra, which is basic and connected. Suppose that A is a right Ae-module of τAe-period 1. (1) If A is an algebra whose Jacobson radical square is zero, then [Formula: see text]. (2) If A is an algebra whose Jacobson radical square is not zero, then [Formula: see text] for some positive integer n.

Finiteness of the strong global dimension of radical square zero algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.

Controlled Wild Algebras

The controlled wild algebra is introduced, a covering criterion for a finite-dimensional algebra to be controlled wild is given, and this criterion is applied to the algebras with radical square zero, algebras with zero relations, local algebras and finite p-group algebras. 2000 Mathematical Subject Classification: 16G20, 16G60.

Strictly wild algebras with radical square zero

It is proved that a wild algebra with radical square zero is strictly wild if and only if it has a wild hereditary algebra as its factor algebra, which, on the one hand supports one conjecture in [3], on the other hand can be used to construct many non-strictly wild algebras.

Postprojective components for algebras with certain heredity conditions

Let \( {\mit\Lambda} \) be a connected artin algebra with the postprojective partition \( {\cal P}_0 , \cdots , {\cal P}_n , \cdots , {\cal P}\!_{\infty } \), and assume that \( {\cal P}_0 \cup {\cal P}_1 \) is closed under indecomposable submodules. Then either \( {\mit\Lambda} \) is the radical square zero algebra given by the quiver \( {\tilde {\bf\it A}}_n \) with cyclic orientation, or the Auslander-Reiten-quiver of \( {\mit\Lambda} \) has a postprojective component \( {\mit\Gamma} \) cosisting of the modules in \( \bigcup \limits _{n\in {\Bbb N}_0}{\cal P}_n \).

Module categories with infinite radical square zero are of finite type

It is well known that an artin algebra A is of finite representation type if and only if red(modA)=0. In this note we deepen this result by showing that(rad(modA))2=0 implies that A is of finite representation type.

OnS-duo rings

A unital left R-module R M is said to have property (S) if every surjective endomorphism of R M is an automorphism, the ring R is called left (right) S-ring if every left (right) R-module with property (S) is Noetherian, R is called S-ring if it is both a left and a right S-ring. In this note we show that a duo ring is a left S-ring if and only if it is left Artinian left principal ideal ring. To do this we shall construct on every non distributive Artinian local ring with radical square zero a non-finitely generated module with property (S). And we give an example of left duo left Artinian left principal ideal ring which is not a left S-ring, showing the necessity of the ring to be duo in the above result.

Commutative rings for which every continuous module is quasi-injective

1. Introduction. The concept of continuous modules owes its origin to Von Neumann's work on continuous geometries [16], and Utumi's work on continuous rings ([12], [13], [14]). L. Jeremy in [3] defined continuous modules, and others ([2], [5], [6], [7], [10]) later provided many interesting properties of such modules. A module is called continuous if (i) every submodule is essential in a summand and (ii) every submodule, isomorphic to a summand, is itself a summand. It can be easily seen, that while every quasi-injective module is continuous ([3], [5], [10]), the converse is not true. It is, therefore, natural to ask: For what classes of rings, is every continuous module quasi-injective? We provide here some classes of commutative rings, for which this is true. Our study is motivated by a general interest in identifying rings for which continuity is a non-trivial concept, and specifically, because only for such rings the existence of continuous hulls [7] poses a problem. We show that large classes of commutative rings, such as noetherian ones, and semiprimary ones with Jacobson radical square zero, have the property that every continuous module is quasi-injective. We give a necessary and sufficient condition, over an arbitrary commutative ring, for every uniform continuous module to be quasi-injective. We also provide necessary and sufficient conditions for some special classes of commutative rings, such as perfect rings, finitary rings, and chain rings, to have the property that every continuous module is quasi-injective. All our rings are commutative, have identity elements, and all modules are unitary rightmodules. Homomorphisms and endomorphisms act on the left. E(M)= EM denotes the injective hull of a module M. A c'M means that A is an essential submodule of M, a "summand" means a "direct summand", and e ~ stands for the annihilator in R of an element e of an R-module. A module A is said to be B-injective if every homomorphism from a submodule of B to A can be extended to B. We recall that a module M is called quasi-continuous (or 7r-injective) if M is closed under all idempotents of EndoRE(M) ([2], [3]). This holds if and only if every decomposition E(M)-- ~ @ E, induces a decomposition M= ~ @ (M hE,) tel tel ([2], Theorem 1.1). Every continuous module is quasi-continuous, and every quasi-continuous module satisfies the condition (i) of the definition of a continuous module. Let Mbe a module, with an injective hull E, then a continuous overmodule C of Mis called a continuous hull of M if Mc C ~ E, and if C c X holds for every continuous module M= X c E. We mention that this definition is independent of the choice of E [7].

Socle reductions and socle projective modules

A useful and simple idea in determining the representation type of an artinian ring A is to relate its module category mod(A) with a module category of another ring A’ whose representation type is known or it is easier to study. This was successfully used by several authors. Let us mention the Coxeter functors connection [9, 10,3,38], stable equivalences [2, 231, the connection between radical square zero rings and hereditary rings [ 11, 221, connections between representations of algebras and representations of partially ordered sets [ 19, 20, 24, 253, or more generally with representations of BOWS [29], and the covering technique [ 131. An idea close to the one above is to find some subcategory 9 of mod(A ) whose representations type is known and to look at the factor category mod(A)/[W] of mod(A) modulo the two-sided ideal (I&?] in mod(A) generated by maps in SK If mod(A)/[SY] Zmod(A’)/[8] and the representation type of A’ is known then we get a lot of information about mod(A) [2,22,23]. On the other hand there are several situations when mod(~)/[~] is equivalent to a nice subcategory 55’ in some mod(R) and there are constructive methods for studying indecomposable objects in V. It follows from [6-S, 31, 32, 34, 361 that the matrix problems technique and the differentiation procedure [ 19, 201 lead to this scheme. In connection with these ideas the notion of a right peak ring R is introduced and socle projective modules mod,,(R) over R are studied in [31,32f. It is shown there that several factor categories of mod(A), where A is an artinian ring, are of the form mod,,(R), where R is a right peak ring. In particular, if A = ( t “y.‘), where F is a division PI-ring and T is a

Rings with dual continuous right ideals

AbstractIn this paper the structure of rings with dual continuous right ideals is discussed. The main result is the following: If R is a ring with (Jacobson) radical nil, and all of its finitely generated right ideals are dual continuous, then where S is a finite direct sum of local rings each of which has its radical square zero, or is a right valuation ring, T is semiprimary right semihereditary ring, and M is an (S, T)-bimodule such that all of its finitely generated T-submodules are projective. A partial converse of this result is obtained: any matrix ring of the above type with M = 0 has all of its finitely generated right ideals dual continuous.

Representation Theory of Algebras Stably Equivalent to an Hereditary Artin Algebra

Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.

Representation theory of algebras stably equivalent to an hereditary Artin algebra

Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.

Stable Equivalence for Some Categories With Radical Square Zero

Stable Equivalence for Some Categories With Radical Square Zero

Stable equivalence for some categories with radical square zero

For certain abelian categories with radical square zero, containing artin rings with radical square zero as a special case, we give a way of constructing hereditary abelian categories stably equivalent to them, i.e. such that their categories modulo projectives are equivalent categories. Introduction. Let C be an additive category where idempotents split. Denote as in [2] by Mod C the category of contravariant additive functors from C to abelian groups (called C-modules), and by mod C the full subcategory of finitely presented functors. We shall further assume that each object in C is a finite direct sum of indecomiposable objects in C, and that the endomorphism ring Rc of an indecomposable object C is a division ring modulo its radical, so that a decomposition of an object in C into a finite direct sum of indecomposable objects is unique. Assume also that D = mod C is abelian and that the simple C-modules are finitely presented. All the above conditions on C and D = mod C will be assumed throughout the paper. Important cases where the above assumptions hold are D = mod A, the category of finitely generated (left) modules over a left artin ring A, and the case where D is a dualizing R-variety for a commutative artin ring R (see [4] for definition). Let D/P denote the category D modulo projectives (see [4]). We say that D = mod C and D' = mod C' are (projectively) stably equivalent if D/P and D'/P are equivalent categories. For dualizing R-varieties this coincides with our previous definition of stable equivalence [4]. In [4] we discussed Loewy length for dualizing R-varieties D = mod C, denoted by LL(D), coinciding with the ordinary Loewy length in the case of rings. We showed in [5] that if D = mod C is a dualizing R-variety with LL(D) < 2, there is a hereditary dualizing R-variety stably equivalent to D. In this paper we shall obtain this result in a completely different way. Our method here is interesting because we at the same time get a direct way of constructing a hereditary dualizing R-variety stably equivalent to a dualizing Received by the editors August 16, 1974. AMS (MOS) subject classifications (1970). Primary 16A46, 16A62; Secondary 18A25, 18E10. Copyright i 1975. American Mathematical Society

Socle conditions for QF− 1 rings

Let R be an associative ring with 1. If RM is a left iϊNmodule, then M can be considered as a right ^-module, where ^ = Horn (RM, RM) is the centralizer of RM. There is a canonic ring homomorphism p from R into the double centralizer 2& = Horn (Λf , M-&) of RM. For a faithful module RM, the homomorphism /> is injective, and RM is called balanced (or to satisfy the double centralizer condition) if p is surjective. An artinian ring R is called a QF-1 ring if every finitely generated faithful ϋί-module is balanced. This definition was introduced by R. M. Thrall as a generalization of quasi-Frobenius rings, and he asked for an internal characterization of QF-1 rings. The paper establishes three properties of QF-1 rings which involve the left socle and the right socle of the ring; in particular, it is shown that QF-1 rings are very similar to QF-S rings. The socle conditions are necessary and sufficient for a (finite dimensional) algebra with radical square zero to be QF-19 and thus give an internal characterization of such QF-1 algebras. Also, as a consequence of the socle conditions, D. R. Floyd's conjecture concerning the number of indecomposable finitely generated faithful modules over a QF-1 algebra is verified. In fact, a QF-1 algebra has at most one indecomposable finitely generated faithful module, and, in this case, is a quasi-Frobenius algebra.