Von Neumann finite endomorphism rings

Let K be a field of characteristic zero, G a group acting on a nonempty set X and KX the permutation module induced by this action. By studying traces of idempotents, we prove that the endomorphism ring End_K[G](KX) is von Neumann finite under certain conditions for the action of G on X. This generalizes a classical result by Kaplansky for the group ring of G over K.     

Monomial modules and graded Betti numbers

Let K be a field, S = K[x ₁,…, x _n ], the polynomial ring over K, and let F be a finitely generated graded free S-module with homogeneous basis. Certain invariants, such as the Castelnuovo-Mumford regularity and the graded Betti numbers of submodules of F, are studied; in particular, the componentwise linear submodules of F are characterized in terms of their graded Betti numbers.     

Links ore sets,and classical localizations

The notion of a simple ring D_Gderived from a group ring KG is introduced in case K is a field and G is an infinite residually finite group. The close link between D_Gand K_G is exploited in both directions: first, for a simple proof of the Kaplansky's conjecture concerning direct finiteness of KG. Second, to show that D_Gprovides counter-examples to some conjectures dealing with von Neumann regular rings and the rings all of whose one-sided ideals are generated by idempotents.     

Algebras whose right nucleus is a central simple algebra

We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D, where F is algebraically closed in K.We then give a short direct proof that every p-algebra of degree m, which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic $p>0$ whose right nucleus is a division p-algebra.     

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.     

Some algebraic characterizations of F-frames

In pointfree topology, F-frames have been defined by Ball and Walters-Wayland by means of a frame-theoretic translation of the topological characterization of F-spaces as those whose cozero-sets are C*-embedded. This is a departure from the way in which F-spaces were defined by Gillman and Henriksen as those spaces X for which the ring C(X) is Bézout, meaning that every finitely generated ideal is principal. In this note, we show that, as in the case of spaces, a frame L is an F-frame precisely when the ring ${\mathcal{R}L}$ of continuous real-valued functions on L is Bézout. A commutative ring with identity is called almost weak Baer if the annihilator of each element is generated by idempotents. We establish that ${\mathcal{R}L}$ is almost weak Baer iff L is a strongly zero-dimensional F-frame.     

On group rings of linear groups

Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number $N=N(H)$. We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field.     

Abhyankar places admit local uniformization in any characteristic

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F|K, and the extension FP|K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R⊆K of Krull dimension dimR?2, assuming that the valued field (K,vP) is defectless, the factor group vPF/vPK is torsion-free and the extension of residue fields FP|KP is separable. The results also include a form of monomialization.     

On the idempotents,nilpotents, units and zero-divisors of finite rings

In this article, first we find the number of idempotents and the zero-divisors of a matrix ring over a finite field F. Next, given the order of the Jacobson radical of the finite unital ring R, we find the number of units, nilpotents and zero-divisors of R. Moreover, we find an upper bound for the number of idempotents of a finite ring which is in general smaller than the upper bound found by MacHale [Proc. R. Ir. 1982;82A(1):9–12]. Finally, we find the above-mentioned numbers in some matrix rings and quaternion rings.     

On Projectively Fully Transitive Abelian p-Groups

There are two natural questions which arise in connection with the endomorphism ring of an Abelian group: when is the ring generated by its idempotents and when is the ring generated additively by its idempotents? The present work investigates these two questions for Abelian p-groups. This leads in a natural way to consideration of two strengthened versions of Kaplansky’s notion of full transitivity, which we call projective full transitivity and strong projective full transitivity. We establish, inter alia, that these concepts are strictly stronger than the classical concept of full transitivity but there are nonetheless many strong parallels between the notions.     

Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.     

Cube root Ramanujan formulas and elementary Galois theory

The cube root Ramanujan formulas are explained from the point of view of Galois theory. Let F be a cyclic cubic extension of a field K. It is proved that the normal closure over K of a pure cubic extension of F contains a certain pure cubic extension of K. The proposed proof can be generalized to radicals of any prime degree q. In the case where the base field K is the field of rational numbers and the field F is embedded in the cyclotomic extension obtained by adding the pth roots of unity, the corresponding simple radical extension of the field of rational numbers is explicitly constructed. The proof of the main result illustrates Hilbert’s Theorem 90. An example of a particular formula generalizing Ramanujan’s formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the field F is given.     

Generalized model matching and (F,G)-invariant submodules for linear systems over rings

A generalized version of the exact model matching problem (GEMMP) is considered for linear multivariable systems over an arbitrary commutative ring K with identity. Reduced forms of this problem are introduced, and a characterization of all solutions and minimal order solutions is given, both with and without the properness constraint on the solutions, in terms of linear equations over K and K-modules. An approach to the characterization of all stable solutions is presented which, under a certain Bezout condition and a freeness condition, provides a parametrization of all stable solutions. The results provide an explicit parametrization of all solutions and all stable solutions in case K is a field, without the Bezout condition. This is achieved through a very simple characterization and a generalization to an arbitrary field K of the “fixed poles” of the model matching problem in terms of invariant factors of a certain polynomial matrix. The results also show that whenever the GEMMP has a solution, there exist solutions whose poles can be chosen arbitrarily as far as they contain the “fixed poles” with the right multiplicities (in the algebraic closure of K). Implications of these results in regard to inverse systems are shown. Equivalent simpler forms (in state space form) of the problem are shown to be obtainable. A theory of finitely generated (F,G)-invariant submodules for linear systems over rings is developed, and the geometric equivalent of the model matching problem—the dynamic cover problem—is formulated, to which the results of the previous sections provide a solution in the reduced case.     

The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

Simultaneous orderings in function fields

For every Dedekind domain R, Bhargava defined the factorials of a subset S of R by introducing the notion of p-ordering of S, for every maximal ideal p of R. We study the existence of simultaneous ordering in the case S=R=O_K, where O_K is the ring of integers of a function field K over a finite field F_q. We show, that when O_K is the ring of integers of an imaginary quadratic extension K of F_q(T), K=F_q(T)/(Y²-D(T)), then there exists a simultaneous ordering if and only if degD?1.     

Adelic openness for Drinfeld modules in generic characteristic

Let φ be a Drinfeld A-module of arbitrary rank and generic characteristic over a finitely generated field K. If the endomorphism ring of φ over an algebraic closure of K is equal to A, we prove that the image of the adelic Galois representation associated to φ is open.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

On cubic Galois field extensions

We study Morton's characterization of cubic Galois extensions F/K by a generic polynomial depending on a single parameter s∈K. We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F. For K=Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.     

Valuation rings and simple transcendental field extensions

If a valuation ring V on a simple transcendental field extension K₀(X) is such that the residue field k of V is not algebraic over the residue field k₀ of V₀=V∩K₀, then for k₀ a perfect field it is shown that k is obtained from k₀ by a finite algebraic followed by a simple transcendental field extension.