CLASSICAL QUOTIENT RINGS OF GROUP RINGS

Abstract

Throughout G will denote a free Abelian group and Z(R) the right singular ideal of a ring R. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R/Z(R) is semiprime, Z(R) is rationally closed, and Z(R) contains no closed uniform right ideals. We prove that R is a Cl-ring if and only if the group ring RG is a C1-ring. If RG has the additional property that bRG is dense whenever b is a right nonzero-divisor, then the complete ring of quotients of RG is a classical ring of quotients.     

ON ANTISIMPLE GAMMA RINGS

ABSTRACT

In this note, we define the antisimple radical, A(M), of a Γ-ring M. A(M) is shown to be a special radical, and two characterizations of antisimple rings due to Szész are extended to Γ-rings. If R is the right operator ring of M, then A(R)* = A(M), where A(R) is the antisimple radical of R. If m,n are positive integers, then A(M_mn) = (A(M))_mn, where M_mn denotes the group m x n matrices over M, considered as a Γ_nm -ring with the operations of matrix addition and multiplication.     

ON SOME PROPERTIES OF IF RINGS

Abstract

We show that left IF rings (rings such that every injective left module is flat) have certain regular-like properties. For instance, we prove that every left IF reduced ring is strongly regular. We also give characterizations of (left and right) IF rings. In particular, we show that a ring R is IF if and only if every finitely generated left (and right) ideal is the annihilator of a finite subset of R.     

The Graded Preprojective Algebra of a Quiver

The paper introduces a new grading on the preprojective algebraof an arbitrary locally finite quiver. Viewing the algebra asa left module over the path algebra, the author uses the gradingto give an explicit geometric construction of a canonical collectionof exact sequences of its submodules. If a vertex of the quiveris a source, the above submodules behave nicely with respectto the corresponding reflection functor. It follows that whenthe quiver is finite and without oriented cycles, the canonicalexact sequences are the almost split sequences with preprojectiveterms, and the indecomposable direct summands of the submodulesare the non-isomorphic indecomposable preprojective modules.The proof extends that given by Gelfand and Ponomarev in thecase when the finite quiver is a tree. 2000 Mathematics SubjectClassification 16G10, 16G70.     

BGP-reflection functors and cluster combinatorics

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections” on the set of almost positive roots Φ_≥−1 associated with a finite dimensional semi-simple Lie algebra. Combining this with the tilting theory in cluster categories developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054], we give a unified interpretation via quiver representations for the generalized associahedra associated with the root systems of all Dynkin types (simply laced or non-simply laced). This confirms the Conjecture 9.1 in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054] for all Dynkin types.     

UNITS AND ISOMORPHISM IN GROUP RINGS

Abstract

In this brief note, we are concerned with two problems. First we study the units of RG when R is commutative and G is right-ordered. Earlier results of the author and also of Karpilovsky are improved. Secondly, we show that if R and S are P. I. D.'s which do not contain fields and R>x< = S>x<, then R = S.     

Linearity of stability conditions

Abstract

For modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear.     

ON SOME SPECIAL CLASSES OF PRIME RINGS

Abstract

Given a non-zero cardinal α, a ring R is said to be SP(α) if a is the first cardinal for which every non-zero element of R has an insulator of cardinality less than α + 1.

It is shown that the class of SP(α) rings is a special class (in the sense of Andrunakievi?) for each α. A theorem of Groenewald and Heyman (also Desale and Varadarajan) to the effect that the class of all strongly prime rings is a special class is obtained as a corollary. Every SP(α) ring has an SP(α) rational extension ring with identity.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

PRIMITIVITY IN MATRIX NEAR-RINGS

Abstract

The relationships between modules of a near-ring R and the matrix near-ring IM_n(R) are studied, especially as regards primitivity. It is shown that R is 2-primitive iff IM_n(R) is.     

CHARACTERIZING THE NIL RADICAL OF A GAMMA RING

ABSTRACT

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.     

Gorenstein quivers

In this paper we characterize when the path ring associated to a quiver is Gorenstein (in the sense of Iwanaga [9]). Then, by using the notion of a Gorenstein category (cf. [2]), we extend the classes of quivers whose corresponding category of representations has finite Gorenstein global dimension. This extension includes non-noetherian quivers. E. E., S.E., and J.R.G.R., partially supported by the DGI MTM2005-03227. Estrada’s work was supported by a MEC/Fulbright grant from the Spanish Secretaría de Estado de Universidades e Investigación del Ministerio de Educación y Ciencia. Received: 28 February 2006     

COPURE INJECTIVE MODULES

Abstract

A module is said to be copure injective if it is injective with respect to all modules A ? B with B/A injective. We first characterize submodules that have the extension property with respect to copure injective modules. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules. Finally, we show that the quotient categories of reduced copure injective modules and reduced h- divisible modules are isomorphic.     

Generic variables in acyclic cluster algebras

Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra A(Q). We prove that the set G(Q) of generic variables contains naturally the set M(Q) of cluster monomials in A(Q) and that these two sets coincide if and only if Q is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig’s dual semicanonical basis. This allows to compute explicitly the generic variables when Q is a quiver of affine type. When Q is the Kronecker quiver, the set G(Q) is a Z-basis of A(Q) and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.     

TWO CLASSES OF HOMOMORPHISMS IN MODULE CATEGORIES

Abstract

In a category R-Mod a homomorphism α:A → B is called projective if α factors through every epimorphism with B as image. Injective homomorphisms are defined dually. Some properties of such homomorphisms are derived, and it is shown that the hereditariness of the ring R is equivalent to some conditions which can be simply stated in terms of projective and injective homomorphisms.     

Admissible sequences and the preprojective component of a quiver

This paper concerns indecomposable preprojective modules over the path algebra of a finite connected quiver without oriented cycles. For each such module, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the module. An efficient way to compute the module is to recover it from its shortest (+)-admissible sequence. The set of equivalence classes of the above sequences has a natural structure of a partially ordered set. For a large class of quivers, the Hasse diagram of the partially ordered set is isomorphic to the preprojective component of the Auslander-Reiten quiver. The techniques of (+)-admissible sequences yield a new result about slices in the preprojective component.     

STRONGLY PRIME GROUP RINGS

Abstract

A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note is to determine some more group rings which are SP.

For a ring R we also define the strongly prime radical s(R). We then show that s(R)G = s(W) for certain classes of groups.     

Strongly Minimal Modules Over Group Rings

ABSTRACT

We consider modules over a group ring RG where R is a countable Dedekind domain and G is a finite group. We describe the internal structure of those RG-modules which are strongly minimal or satisfy other related model theoretic and algebraic minimality conditions.     

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.     

The Natural Quiver of an Artinian Algebra

The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing upon the associated basic algebra. In Li (J Aust Math Soc 83:385–416, 2007), the notion of the natural quiver of an artinian algebra was introduced and then was used to generalize the Gabriel theorem for non-basic artinian algebras splitting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicals via pseudo path algebras and generalized path algebras respectively. In this paper, firstly we consider the relationship between the natural quiver and the ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabriel theorem is obtained for radical-graded artinian algebras. Moreover, Gabriel-type algebras are introduced to outline those artinian algebras satisfying the generalized Gabriel theorem here and in Li (J Aust Math Soc 83:385–416, 2007). For such algebras, the uniqueness of the related generalized path algebra and quiver holds up to isomorphism in the case when the ideal is admissible. For an artinian algebra, there are two basic algebras, the first is that associated to the algebra itself; the second is that associated to the correspondent generalized path algebra. In the final part, it is shown that for a Gabriel-type artinian algebra, the first basic algebra is a quotient of the second basic algebra. In the end, we give an example of a skew group algebra in which the relation between the natural quiver and the ordinary quiver is discussed.