Endomorphisms of matrix semigroups over division rings

Let be an arbitrary division ring and M _n ( ) the multiplicative semigroup of all n × n matrices over . We describe the general form of endomorphisms of M _n ( ). Supported in part by a grant from the Ministry of Science of Slovenia.     

A duality theorem for hope module algebras over dedekind rings

Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p≧3, which has been proved to be generalized restricted in the sense of [Shul, Shu2]. For a generalized restricted L-module M, the homological support variety ‖L‖_M is defined to be that of the primitive p-envelope P{L). A realization L of P(L) is given in Der(&;(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(Tx)V(p)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].     

Homomorphisms of matrix semigroups over division rings from dimension two to four

Let D be an arbitrary division ring and M_n(D) the multiplicative semigroup of all n×n matrices over D. We study non-degenerate, injective homomorphisms from M₂(D) to M₄(D). In particular, we present a structural result for the case when D is the ring of quaternions.     

Automorphisms of linear semigroups over division rings

Given an-dimensional right vector spaceV over a division ring we denote byS the semigroup of the endomorphisms ofV and designate this semigroup as alinear semigroup. First we prove that every automorphism ofS can be written asT→fTf ⁻¹, wheref∶V→V is a semilinear homeomorphism. Furthermore, we show that every isomorphism between maximal compact subsemigroups ofS is also of this type.     

Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K ^m ;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given. Dedicated to the memory of Professor K G Ramanathan     

On the Galois cohomology of dedekind rings

Let R be a Dedekind domain, G a finite group of automorphisms of R, and A an ambiguous ideal of R i.e., σA = A for all σ ∈ G. The Tate groups Hⁿ(G, A) are considered as R^G-modules. A localization theorem is proved and the precise R^G-module structure determined in a particular case. In addition some remarks are made concerning cohomological triviality.     

Modules over formal matrix rings

This work contains some new and known results on modules over formal matrix rings. The main results are presented with proofs.     

Strongly clean matrix rings over commutative local rings

We will completely characterize the commutative local rings for which M_n(R) is strongly clean, in terms of factorization in R[t]. We also obtain similar elementwise results which show additionally that for any monic polynomial f∈R[t], the strong cleanness of the companion matrix of f is equivalent to the strong cleanness of all matrices with characteristic polynomial f.     

Endomorphisms of semigroups of invertible nonnegative matrices over ordered rings

Let R be a linearly ordered commutative ring with 1/2 and G _n (R) be the subsemigroup of GL _n (R) consisting of matrices with nonnegative elements. In this paper, we describe endomorphisms of this semigroup for n ≥ 3.     

Simple modules over generalized matrix rings

We study simple modules over the ring of generalized matrices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 245–247, 2007.     

Elementary matrix reduction over J-stable rings

A commutative ring R is J-stable provided that R∕aR has stable range 1 for all a?J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3 Gillman, L., Henriksen, M. (1956). Some remarks about elementary divisor rings. Trans. Amer. Math. Soc. 82:362–365.[Crossref] , [Google Scholar], Theorem 8], [4 Larsen, M., Lewis, W., Shores, T. (1974). Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187:231–248.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 4.1], [7 McGovern, W. W. (2008). Bézout rings with almost stable range 1. J. Pure Appl. Algebra 212:340–348.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.7], [8 Moore, M. E. (1975). A strongly complement property of Dedekind domain. Czechoslovak Math. J. 25(100):282–283. [Google Scholar], Theorem], [9 Moore, M., Steger, A. (1971). Some results on completability in commutative rings. Pacific J. Math. 37:453–460.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.1], [14 Zabavsky, B. V. (1996). Generalized adequate rings. Ukrainian Math. J. 48:614–617.[Crossref] , [Google Scholar], Theorem 1] and [18 Zabavsky, B. V., Komarnyts’kyi, M. Y. (2010). Cohn-type theorem for adequacy and elementary divisor rings. J. Math. Sci. 167:107–111.[Crossref] , [Google Scholar], Theorem 7].     

Elementary matrix reduction over certain rings

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR+bR = R??x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.     

Injective modules over formal matrix rings

We describe injective modules over formal matrix rings.