Schur rings over a Galois ring of odd characteristic

It is proved that any Schur ring over a Galois ring of odd characteristic is either normal, or of rank 2, or a non-trivial generalized wreath product. The normal Schur rings are characterized as a special subclass of the cyclotomic Schur rings.     

On schur rings over cyclic groups

In this paper, we introduce the notion of wedge product of Schur rings. We show that for any nontrivial Schur ringS over a cyclic groupG, if there is a subgroupH such that Σ _g ε _H g Σ _g∈H g ∉S, thenS is either a dot product or wedge product for some Schur rings over smaller cyclic groups.     

Schur rings and non-symmetric association schemes on 64 vertices

In this paper we enumerate essentially all non-symmetric association schemes with three classes, less than 96 vertices and with a regular group of automorphisms. The enumeration is based on a computer search in Schur rings. The most interesting cases have 64 vertices.In one primitive case and in one imprimitive case where no association scheme was previously known we find several new association schemes. In one other imprimitive case with 64 vertices we find association schemes with an automorphism group of rank 4, which was previously assumed not to be possible.     

Primitive Idempotents of Schur Rings

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring S, S contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in S. We show that if S is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in S. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored.     

Zariski extensions and biregular rings

In this paper we study the structure of Zariski central rings with regular center i.p. biregular rings, and we obtain structure theorems for algebras which are finitely generated over their regular center, etc. Characterizations of certain classes of rings are being obtained by using localization at prime ideals and local-global theorems.     

Local rings of exchange rings

We characterize the exchange property for non-unital rings in terms of their local rings at elements,and we use this characterization to show that the exchange property is Morita invariant for idempotent rings.We also prove that every ring contains a greatest exchange idela(with respect to the inclusion).     

Isomorphisms of formal matrix incidence rings

Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.     

MDS and self-dual codes over rings

In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings. We also define the Reed–Solomon codes over principal ideal rings.     

Hacque-Amitsur rings

We consider a class of rings that combines classes from a theorem of Amitsur and the classes characterized by Hacque. A structure theorem is proved, and we use it to describe left nonsingular Amitsur rings with a finite left Goldie dimension.Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 190–210, March-April, 1995.     

Bases for rings of coinvariants

We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada     

On GC2 modules and their endomorphism rings

Let R be a ring. A module M_R is said to be GC₂ if for any N≤ M with N? M, N is a direct summand of M. In this article, we give some characterizations and properties of GC₂ modules and their endomorphism rings, and many results on C ₂ modules and GC₂ rings are generalized to GC₂ modules.     

The structure of commutative rings with monomorphic flat envelopes

It is obvious that OF and Von Neumann regular rings have monomorphic flat envelopes. In this paper we completely describe the structure,in terms of OF and Von Neumann regular rings, of those commutative rings all of whose modules have a monomorphic flat envelope (m.f.e. ). For that, we introduce the notion of locally QF ring with m.f.e., whose structure is given in terms of OF rings. It turns out that a commutative ring R with m.f.e. is characterized as a (essential) subdirect product of a locally QF ring with m.f.e. and a Von Neumann regular ring, with the latter flat as an R-module.     

Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

Intermediate rings between matrix rings and Ornstein dual pairs

We show that isomorphism of intermediate rings between row and column finite matrix rings and row finite matrix rings implies Morita equivalence of the coefficient rings and equality of the cardinality of the set of indices. Among the applications we extend the Isomorphism Theorem for Dual Pairs over Division Rings to Ornstein dual pairs over any class of rings for which Morita equivalence implies isomorphism.     

Decomposition of group rings and the converse of Schur’s lemma

In this work, we give necessary and su?cient conditions for a group ring (of finite group) to satisfy the converse of Schur’s lemma for group rings of finite groups. Rings considered here are commutative or noncommutative perfect. Some cases of twisted group rings are studied. Also we introduce the notion of semi-CSL ring.     

Multiserial rings

We introduce the notion of multiserial (n-serial) rings and study their properties. The second-order minors of such rings are investigated. We also find all possible forms of quivers for Noetherian and hereditaryn-serial rings and describe the structure of semiprime and hereditaryn-serial rings. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1223–1235, September, 1996.     

On cubic rings and quaternion rings

In this paper, we show that the orbits of some simple group actions parametrize cubic rings and quaternion rings.     

On automorphisms of groups,rings, and algebras

We consider the structure of groups and algebras that can be represented as automorphisms, respectively derivations, of bilinear maps. Representations of that sort arise when we attempt to describe the automorphisms of groups, rings, and algebras that are nilpotent. We introduce exact sequences that capture structure and prove theorems of Morita and Skolem–Noether type. We apply these results to compute automorphisms of groups and rings.     

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasi-projective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky’s theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz’s related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.     

On structure sheaves of non-commutative rings

In this paper, we shall present a general process for structure presheaves of general rings which are not necessarily left noetherian nor commutative. We shall give several necessary and sufficient conditions such that the structure presheaves are actually sheaves. These results generalize considerably Grothendieck’s original one. as well as various existing corresponding results.