The structure of Johns rings

This paper studies the question, if $R$ is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over $R$? Theorem 2.6. If $R$ is an algebra over a field with at least $q$ elements, and $R$ satisfies ${x^q} = 0$, and $G$ is a group with an abelian subgroup of index $k$, then the group ring $R(G)$ satisfies ${x^t} = 0$, where $t = q{k^2} + 2$. Theorem 3.2. If $R$ is a ring satisfying a standard identity, and $G$ is a finite group, then $R(G)$ satisfies a standard identity. Theorem 3.4. If $R$ is an algebra over a field, and $R$ satisfies a standard identity, then the $k$-by-$k$ matrix ring ${R_k}$ satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.

It is shown that if F is a field of characteristic zero and G is a group such that the group ring F[G] is semilocal then G must be finite.A generalization to group rings over rings is given.A ring R is semilocal if R/J(R) is artinian, where I(R) denotes the Jacobson radical of R. R is said to be local if R/I(R) is a division ring.It is well known that the group ring is never local for a field of characteristic zero unless the group is trivial.As it is conjectured that J(F[G]) = (0) whenever Fis a field of characteristic zero, we expect F[G] semilocal to imply G finite, and this is easily proved if F is not algebraic over the rationals, by a theorem of Amitsur [1].The result in this paper may be interpreted as saying that the radical of the rational group ring cannot be "too large".We would like to thank D. S. Passman for his helpful suggestions.Lemma 1.Let K be a central subfield of a division ring D and let T G Mn(D), the full n by n matrix ring over D. Then the set S (T) = {k G K: 1 -k~ T is singular] has at most n elements.Proof.When written on the right, the elements of Mn(D) may be regarded as left Z)-linear transformations from the vector space D" to D".If 1 -k~] T is singular, there is a nonzero vector v G D" such that v(\ -k~ T) = 0, or vT = kv since k commutes with v. Hence k is an eigenvalue of T. Standard arguments of linear algebra show that eigenvectors in D" corresponding to distinct eigenvalues of T in the centre of D are Z)-linearly independent. Corollary.Let R be a completely reducible K-algebra and let x G R. Then the set S(x) = {k G K: 1 -k~ x is not a unit in R] is finite.The following clever lemma forms a major part of the proof of the theorem in [3],Lemma 2 (Formanek).Let K be a subfield of the reals and let x = 2"i aS

A ring R is called right johns if R is a right noetherian ring such that every right ideal A of R is an annihilator (i.e. rl (A) = A for all right ideals A of R). R is called strongly right johns if every matrix ring over R is right johns. In this paper it is shown that every right johns ring is a cogenerator and hence quasi-frobenius. By using this result we show that every strongly right johns ring is quasi-frobenius without any extra condition. Mathematics Subject Classification: 16D50, 16L60

Localization in a Morita context with applications to fixed rings

The theory of Noetherian rings may be said to have begun with Goldie's 1958 paper. Since then the theory has steadily gathered strength. By now, various methods and results from the theory of commutative Noetherian rings have been adapted to non-commutative Noetherian rings. The theory has also contributed to and benefitted from developments in three nearby areas: P.I. rings, enveloping algebras of Lie algebras, and group rings of polycyclic-by-finite groups.

For each integer n ≥ 2 , simple noetherian hereditary rings R are constructed such that R has uniform (Goldie) dimension n but R is: not isomorphic to a matrix ring over an integral domain. That R is not a matrix ring is verified K-theoretically, using Quillen’s machinery [3] to compute K0(R). In fact, for these examples K0(R) is free abelian of rank n.

A new construction method for codes using encodings from group rings is presented. They consist primarily of two types, zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; e.g. cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. Using an isomorphism between group rings and a certain well- defined ring of matrices, equivalent matrix codes are established with resulting generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties may more easily be expressed in terms of group ring properties.

The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the rank of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

Nilpotent decomposition in integral group rings

The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report. The main highlights of the conference were: The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio. Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups. the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units. discussion of the present state of art via several survey talks presented and the problem sessions The group G determines its integral group ring \mathbb Z G and its group V(\mathbb Z G) of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.

definitions are recalled in the section of preliminaries in this paper. In this paper we compute the ring of differential operators for some noncommutative rings, namely the Axumaya algebras and the Heisenberg algebras. The initial interest was to compute these rings for matrix algebras and the Weyl algebras, which were easily generalized. We consider Azumaya algebras over a noetherian ring. We conclude that the ring of differential operators are generated as modules by the ring of differential operators on their centre and homomorphisms given by multiplication by elements of the bigger ring (called inner homomorphisms). That is, if R is the centre of A (where A is an Azumaya algebra over R), we show that Dk(R) can be embedded into Dk(A) and that Dk(A) = (A N R A o ).Dk(R).(A N R A o ), that is, Dk(A) is gener

Modules with chain conditions up to isomorphism

In this paper, the authors describe homomorphisms of Lie groups into the groups u(R) of invertible elements of rings R for a large class of rings R, which contains, in particular, subrings of matrix rings and also Noetherian rings.

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module.

The aim of this thesis is to investigate the circumstances under which group rings over fields have non-zero socle, i.e. contain minimal one-sided ideals. After an introductory chapter, we consider the special case o₊ a periodic abelian group and a non-modular field (that is, a field of characteristic prime to the orders of the elements of the group). This special case, and the background material contained in Chapter III, serve as preparation for our principal results, which concern, locally finite groups. We establish necessary and sufficient conditions on an arbitrary field K and a locally finite group 0 for the group ring KG to contain minimal one-sided ideals: the most important condition is that G should be a Cernikov group. We then examine the structure of KG when these conditions are satisfied. Vie show that KG has a finite series of ideal3 each factor of which i3 a direct sum of quasi-Frobenius rings, and characterize the socle of KG. Me also classify indecomposable KG-modules, and determine (for countable but not necessarily locally finite groups G) necessary and sufficient conditions for all indecomposable KG-modules to be irreducible. In the final chapter we consider non-locally-finite groups, conjecturing that group rings of such groups never contain minimal one-sided ideals. We establish the truth of this conjecture for several classes of groups, and also consider semiartinian group rings.