Showing papers on "Ring (mathematics) published in 1988"

GPIs having coefficients in Utumi quotient rings

Abstract: Let R be a prime ring and let U be its Utumi quotient ring. We prove the following: (1) If R satisfies a GPI having all its coefficients in U, then R satisfies a GPI having all its coefficients in R. (2) R and U satisfy the same GPIs having their coefficients in U. The main purpose of this paper is to generalize the two main theorems on generalized polynomial identities (GPIs) in [7] to their full generality. Our improvement is in two respects: (1) The coefficients of the generalized polynomial identities are allowed to lie in the Utumi quotient rings instead of in the Martindale quotient rings. (2) The assumption that the generalized polynomial identity is multilinear and homogeneous is removed. Our theorems hold for any arbitrary generalized polynomial identity. Results in this generality seem to be interesting and should be useful elsewhere. In what follows, R is always a prime associative ring, not necessarily with identity 1. Let UR be the maximal rational extension of RR (as right R-modules) (see the definition and Theorem 6 on p. 59 [2]). Since R is left faithful (p. 67 [2]), there is a natural ring operation on U which induces the module operation U x R -) U (Proposition F on p. 68 [2]). U endowed with the natural ring operation is an overring of R and is called the Utumi quotient ring of R. The Utumi quotient ring of R can also be characterized axiomatically as follows: A right ideal p of R is said to be rational if and only if RR is a rational extension Of PR (rational right ideals are also called dense right ideals in [6]). The Utumi quotient ring of R is a ring U satisfying the following axioms: (1) R is a subring of U. (2) For each a E U, there exists a rational right ideal p of R such that ap C R. (3) If a E U and ap = 0 for some rational right ideal p of R, then a = 0. (4) For any rational right ideal p and for any right R-module homomorphism 0: PR -RR, there exists a E U such that 0(r) = ar for all r E p. For a prime ring R, a nonzero two-sided ideal is obviously a rational right ideal of R. In the above axioms, if we consider only nonzero two-sided ideals instead of rational right ideals, then we obtain the Martindale quotient ring, which we denote by Q (see [8] for the definition and [5] for the axiomatic formulation). Q can be Received by the editors June 29, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A38; Secondary 16A08, 16A12.

Algebraic Cohomology and Deformation Theory

Abstract: We should state at the outset that the present article, intended primarily as a survey, contains many results which are new and have not appeared elsewhere. Foremost among these is the reduction, in §28, of the formal deformation theory of a smooth compact complex algebraic variety χ to that of a single ring built from χ. Others include the relationship between the classical Hodge decomposition of the cohomology of an analytic manifold and the more recent Hodge decomposition of the cohomology of a commutative algebra, the invariance of the Euler characteristic of an algebra under deformation, the correspondence between the deformation theories for Morita equivalent algebras, much of the work on the deformation of presheaves (diagrams) of algebras, and the explicit description of the (algebraic) Hodge decomposition for regular affine algebras. However, in line with the goals of a survey article, we have tried to maximize the exposition, including details only in so far as they aid in this purpose. Many proofs are sketched; many others, including the most difficult, are omitted altogether.

Model theory and modules

Abstract: The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance It is these aspects that I will discuss in this chapter, although I will make some comments on the model theory of modules per se Our default is that the term “module” will mean (unital) right module over a ring (associative with 1) R The category of these is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the notation R-Mod denotes the category of left R-modules By Ab we mean the category of Abelian groups In Part 1 we introduce the general concepts and in Part 2 we discuss these in more specific contexts References within the text, as well as those in the bibliography, are not comprehensive but are intended to lead the reader to a variety of sources

Computing on an anonymous ring

Abstract: The computational capabilities of a system of n indistinguishable (anonymous) processors arranged on a ring in the synchronous and asynchronous models of distributed computation are analyzed. A precise characterization of the functions that can be computed in this setting is given. It is shown that any of these functions can be computed in O(n2) messages in the asynchronous model. This is also proved to be a lower bound for such elementary functions as AND, SUM, and Orientation. In the synchronous model any computable function can be computed in O(n log n) messages. A ring can be oriented and start synchronized within the same bounds.The main contribution of this paper is a new technique for proving lower bounds in the synchronous model. With this technique tight lower bounds of t(n log n) (for particular n) are proved for XOR, SUM, Orientation, and Start Synchronization. The technique is based on a string-producing mechanism from formal language theory, first introduced by Thue to study square-free words. Two methods for generalizing the synchronous lower bounds to arbitrary ring sizes are presented.

Applications of a new K-theoretic theorem to soluble group rings

Abstract: Let R be a ring and let G be a soluble group. In this situation we shall give necessary and sufficient conditions for RG to have a right Artinian right quotient ring. In the course of this work, we shall also consider the Goldie rank problem for soluble groups and record an affirmative answer to the zero divisor conjecture for soluble groups.

Differential Operators on an Affine Curve

Abstract: Let X denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero. Denote by 3)x the sheaf of differential operators on X, and 2)(X) = F(X, 3)x), the ring of global differential operators on X. The following is established: THEOREM. 3)(X) is a finitely generated k-algebra, and a noetherian ring. Furthermore, 2)(X) has a unique minimal non-zero ideal J, and 3)(X)/J is a finite-dimensional k-algebra. Let X denoted the normalisation of X, and n: X—*X the projection map. The main technique is to compare 2)(X) with 3)(X).

Ordinal numbers and the Hilbert basis theorem

Abstract: In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates x1,…,xm. The Hilbert basis theorem asserts that for all K and m, every ideal in the ring K[x1,…,xm] is finitely generated. This theorem is of fundamental importance for invariant theory and for algebraic geometry. There is also a generalization, the Robson basis theorem [11], which makes a similar but more restrictive assertion about the ring K〈x1,…,xm〉 of polynomials over K in mnoncommuting indeterminates.In this paper we study a certain formal version of the Hilbert basis theorem within the language of second order arithmetic. Our main result is that, for any or all countable fields K, our version of the Hilbert basis theorem is equivalent to the assertion that the ordinal number ωω is well ordered. (The equivalence is provable in the weak base theory RCA0.) Thus the ordinal number ωω is a measure of the “intrinsic logical strength” of the Hilbert basis theorem. Such a measure is of interest in reference to the historic controversy surrounding the Hilbert basis theorem's apparent lack of constructive or computational content.

The Radical of the Homotopy Lie-algebra

Abstract: Introduction. Let T be a simply connected CW complex of finite type. Its homotopy Lie algebra is the graded rational Lie algebra 7r*(9 T) 0 Q, equipped with the Samelson product. Let A be a local (noetherian) ring with residue field k (of arbitrary characteristic). Its homotopy Lie algebra, 7r*(A), is the naturally defined graded Lie algebra whose enveloping algebra is ExtA(k, k)-cf. [Av]. These graded Lie algebras have finite type (i.e. are finite dimensional in each degree) and are concentrated in strictly positive degrees. Thus, for the sake of simplicity, we shall use "graded Lie algebra" to mean "graded Lie algebra of finite type concentrated in strictly positive degrees" throughout this paper. On the other hand these graded Lie algebras are usually nonzero in infinitely many degrees. Thus, while we may (and do) form the sum, R, of all the solvable ideals in such a graded Lie algebra and call it the radical, it may well happen that R, itself, is not solvable. Indeed, this does happen in the topological context, since by Quillen's result [Q] every graded rational Lie algebra of finite type occurs as the homotopy Lie algebra of a space. By contrast, in the case of local rings, or of CW complexes with finite Lusternik-Schnirelmann (L.S.) category, serious restrictions apply to the homotopy Lie algebra. Indeed, our first main result is the remarkable

Expository and survey article recent results in hyperring and hyperfield theory

Abstract: This survey article presents some recent results in the theory of hyperfields and hyperrings, algebraic structures for which the "sum" of two elements is a subset of the structure. The results in this paper show that these structures .cannot always be embedded in the decomposition of an ordinary structure (ring or field) in equivalence classes and that the structural results for hyperfields and hyperrings cannot be derived from the corresponding results in field and ring theory.

Performance analysis of slotted ring protocols in HSLANs

Abstract: The performance of a number of slotted-ring protocols supporting integration of synchronous and asynchronous traffic in high-speed local area networks (HSLANs) is evaluated. They are the Cambridge fast ring, a variant of the Cambridge fast ring, and Orwell. The performance of their basic access mechanisms is compared and contrasted with that of the multiple-token ring. The effect of a uniframe scheme for supporting synchronous traffic is examined. A delay analysis of the integrated-services slotted-ring protocols is presented. >

Matrix norms and their applications

Abstract: 1 - Operators in Finite-Dimensional Normed Spaces- 1 Norms of vectors, linear functionals, and linear operators- 2 Survey of spectral theory- 3 Spectral radius- 4 One-parameter groups and semigroups of operators- Appendix Conditioning in general computational problems- 2 - Spectral Properties of Contractions- 1 Contractive operators and isometries- 2 Stability theorems- 3 One-parameter semigroups of contractions and groups of isometries- 4 The boundary spectrum of extremal contractions- 5 Extreme points of the unit ball in the space of operators- 6 Critical exponents- 7 The apparatus of functions on graphs- 8 Combinatorial and spectral properties of l?-contractions- 9 Combinatorial and spectral properties of nonnegative matrices- 10 Finite Markov chains- 11 Nonnegative projectors- 3 - Operator Norms- 1 Ring norms on the algebra of operators in E- 2 Characterization of operator norms- 3 Operator minorants- 4 Suprema of families of operator norms- 5 Ring cross-norms- 6 Orthogonally-invariant norms- 4 - Study of the Order Structure on the Set of Ring Norms- 1 Maximal chains of ring norms- 2 Generalized ring norms- 3 The lattice of subalgebras of the algebra End(E)- 4 Characterization of automorphisms- Brief Comments on the Literature- References

Uniform self-stabilizing rings

Abstract: A self-stabilizing system has the property that it eventually reaches a legitimate configuration when started in any arbitrary configuration. Dijkstra originally introduced the self-stabilization problem and gave several solutions for a ring of processors [Dij74]. His solutions, and others that have appeared, use a distinguished processor in the ring, which can help to drive the system toward stability. Dijkstra observed that a distinguished processor is essential if the number of processors in the ring is composite [Dij82]. We show that there is a self-stabilizing system with no distinguished processor if the size of the ring is prime. Our basic protocol use Θ(n2) states in each processor, where n is the size of the ring. We also give a refined protocol which uses only Θ(n2/ln n) states.

Rings in which every element is the sum of two idempotents

Abstract: Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent.: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x3 = x. (2) If R is a PI-ring in which every element is the sum of two idempotents, then R/P satisfies the identity x3 = x. (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If RRR is quasi-projective, then R is a finite direct sum of copies of GF(2) and/or GF(3).

Local area network with an active star topology comprising ring controllers having ring monitor logic function

Abstract: A star local area network includes a ring bus hub (4) capable of being connected to a plurality of nodes (3, 5, 9) geographically distant from the hub by means of low speed serial links (18, 19, 21, 28). The nodes include processor means (2, 30, 31) for creating messages for transfer on the network. A plurality of duplex communication links (18, 19, 21, 28) connect the nodes to the ring bus hub (4). The hub (40) is comprised of a plurality of ring controllers (10, 12, 14, 16) driven by a common clock source (7). Each ring controller is connected by means of a number of parallel lines to other ring controllers in series to form a closed ring. Each one (3) of the plurality of nodes is geographically distant from the hub (4) and is connected to a corresponding one (10) of the ring controllers by means of one (18, 19) of the duplex communication links. The node controllers including node interface means (40) for transmitting the messages as a contiguous stream of words on the duplex communication link. The ring controllers include ring bus interface means (42) for forming the messages into discrete data packets for insertion onto the ring bus and means (32, 34) for bufferring data messages received form the node and over the ring bus.

Rings with projective socle

Abstract: The class of rings with projective left socle is shown to be closed under the formation of polynomial and power series extensions, direct prod- ucts, and matrix rings. It is proved that a ring R has a projective left socle if and only if the right annihilator of every maximal left ideal is of the form iR, where / is an idempotent in R. This result is used to establish the closure properties above except for matrix rings. To prove this we characterise the rings of the title by the property of having a faithful module with projective socle, and show that if R has such a module, then so does M"(R). In fact we obtain more than Morita invariance. Also an example is given to show that ene, for an idempotent e in a ring R with projective socle, need not have projective socle. The same example shows that the notion is not left-right symmetric.

Elimination theory for the ring of algebraic integers

Abstract: Let 0t be the ring of all algebraic integers, dt c C; its fraction field is an algebraic closure of Q. Note that dt = \\J 0K9 K c C ranging over the finite extensions of Q, and 0K = ring of integers of K. From classfield theory [3], p. 224, one knows that each ideal of &K becomes principal in 0L for some finite (abelian) extension L of X; hence ä? is a (nonnoetherian) Bezout domain, that is, each finitely generated ideal (al5 ..., ak) of 3t is principal.

Computing algebraic formulas with a constant number of registers

Abstract: We show that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by polynomial-length algebraic straight-line programs which use only 3 registers (or 4 registers, depending on some definitions). We also show that polynomial-length products of 3 × 3 matrices compute precisely those functions that polynomial-size formulas compute (whereas, for general rings, polynomial-length 3-register straight-line programs compute strictly more functions than polynomial-size formulas). This can be viewed as an extension of the results of Barrington in [Ba1,Ba2] from the Boolean setting to the algebraic setting of an arbitrary ring.

The Brauer Severi scheme of the trace ring of generic matrices

Abstract: If D is a division algebra with index n with center some field K then the Brauer Severi scheme of D is defined as the scheme representing the left ideals of rank n of D. This definition was subsequently extended to Azumayaalgebras and even to some classes of orders. In the last case the Brauer Severi scheme was defined as a certain connected component of the scheme representing the functor of ideals of minimal rank that are locally split off. See e.g. [Al]. Our aim in this note is to give a general framework for the study of Brauer Severi schemes. We define Bsev n (A, R), for an R-algebra A as the scheme representing the coideals of rank n of A. It is easy to see that the component containing the generic fiber of Bsev n (A, R) → SpecR coincides with the Brauer Severi schemes described above. In the first part of this note we state some general functorial properties of the Brauer Severi scheme. Furthermore we analyze the Brauer Severi scheme of the free algebra over the ground field. This is a nice scheme with an open covering of affine spaces.

Nontrivial uses of trivial rings

Abstract: Four theorems about commutative rings are proved with the aid of the notion of a trivial ring. 0. Introduction. A ring R is trivial if 0 = 1 in R, that is, if R consists of a single element. Although a trivial ring is a boring object, the fact that a construction results in a trivial ring can be quite interesting. In this note we prove the following four theorems from the point of view of trivial rings; in all four theorems R c T are commutative rings with 1. (1) If Rm maps onto R', and m 1. (4) If I is an ideal in R[X] such that 1 E TI, then each element of the annihilator of I n R is nilpotent. Theorem 1 is a standard strong form of the invariance of the rank of a finite-rank free module over a commutative ring [2]; Theorem 2 is a not-so-standard stronger form of the same thing. Alternative formulations presuppose that R is nontrivial and conclude that m > n; I find (1) and (2) more satisfactory. Theorem 1 says that we can derive the equation 0 = 1 from the n equations in R that express the fact that R' maps onto Rn. Theorem 2 says that we can derive the equation 0 = 1 from the conditional equations:

The Local Cohomology Groups of an Affine Semigroup Ring

Abstract: Publisher Summary This chapter provides an overview of the local cohomology groups of an affine semigroup ring. A commutative semigroup ring k [ S ] over a field k is said to be an affine semigroup ring if k [ S ] is an integral domain of finite type over k . This is equivalent to the condition that S is finitely generated and is contained in a free Z-module M of finite rank. An affine semigroup ring k [ S ] has a natural structure of an M-graded ring with respect to the free Z-module M. The dualizing complex and the local cohomology groups are also described in the chapter.

Signal processing system including a bus control module

Abstract: A signal processing system that includes a ring bus; a multiplicity of system modules, each coupled to the ring bus and operative to receive and transfer blocks of data words over the ring bus; and a bus control module coupled to the ring bus and operative to support simultaneous data transfers between specified pairs of system modules in accordance with concurrent execution of multiple programs of data transfer instructions. In an alternate embodiment, the signal processing system may include a plurality of ring buses wherein each system module is coupled to all of the ring buses and operative to receive and transfer blocks of data words over any one of the ring buses. In addition, the bus control module is also coupled to all of the ring buses and operative to support simultaneous data transfers between specified pairs of system modules over all of the ring buses. Moreover, each ring bus may comprise individual bus segments for system module to system module coupling about the ring.

Filtrations and Noetherian symbolic blow-up rings

Abstract: For a one-dimensional prime ideal in a local Noetherian ring it is characterized when the symbolic blow-up ring is an algebra of finite type. More generally, for a filtration of ideals of a local Noetherian ring there is a necessary and sufficient condition for the corresponding Rees ring to be a Noetherian ring. Applications concern asymptotic prime divisors and the analytic spread.

A Cancellation Theorem for Projective Modules over Finitely Generated Rings

Abstract: This chapter reviews various cancellation theorems for projective modules over finitely generated rings. The chapter presents the assumption that P , P ' and Q are projective A-modules such that P Q ≂ P ' Q and rank P ≥ d . Then, ρ ≂ P '. For rank P ≥ d , the group of elementary automorphism of ρ A act transitively on unimodular elements in P A . The chapter also presents the assumption that A is an affine ring of dimension d ≥ 3 over p , then A has projective stable range ≤ d .

Almost jordan rings

Abstract: It is well known that any Jordan ring satisfies the identity: 2((ai)i)i + a((xx)x) = 3(a(xx))x. We show that this identity along with commutativity implies the Jordan identity in any semiprime ring. The proof requires characteristic ^ 2,3. Introduction. A Jordan algebra is a commutative (nonassociative) algebra which satisfies the additional identity ((aa)x)a = (aa)(xa). This identity is called the "Jordan identity". It is not an irreducible identity with respect to commuta- tivity (in the sense of Osborn (3, p. 184)), but is equivalent to the assumption of the following two identities, both of which are irreducible (2, p. 1114).

A proposed method for representing hierarchies

Abstract: A novel diagram (referred to as a ring diagram) helpful in displaying several types of hierarchies is defined. The merits of the diagram are examined and examples of mapping hierarchies taken from the existing literature are provided. >

The differential operator ring of an affine curve

Abstract: The purpose of this paper is to investigate the structure of the ring D(R) of all linear differential operators on the coordinate ring of an affine algebraic variety X (possibly reducible) over a field k (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim X < 1. In this case, it is proved that D(R) is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim X, that the endomorphism ring of any simple (left or right) D(R)-module is finite dimensional over k, that D(R) has a unique smallest ideal L essential as a left or right ideal, and that D(R)/L is finite dimensional over k. The following ring-theoretic tool is developed for use in deriving the above results. Let D be a subalgebra of a left noetherian k-algebra E such that E is finitely generated as a left Dmodule and all simple left E-modules have finite dimensional endomorphism rings (over k), and assume that D contains a left ideal I of E such that E/I has finite length. Then it is proved that D is left noetherian and that the endomorphism ring of any simple left D-module is finite dimensional over k. Introduction. In this paper, we will study the ring D(R) of k-linear differential operators on a commutative k-algebra R, where k is a field of characteristic zero. Of special interest is the case where R is the coordinate ring of an affine algebraic variety X. When X is nonsingular, the ring D(R) has been extensively studied and enjoys many nice properties; for example, D(R) is noetherian. (We will use the term "noetherian" to indicate that a ring is both left and right noetherian.) When X is singular, D(R) need not be noetherian, as shown by J. N. Bernstein, I. M. Gelfand and S. I. Gelfand [3]: if X is the normal cubic cone, i.e., the surface in complex 3-space given by x3 + y3 + Z3 = 0, then D(R) is neither left nor right noetherian. Thus a major goal is to discover for which varieties X the ring D(R) is noetherian. The main contribution of this paper is to prove that D(R) is noetherian when dim X < 1, and to develop some of the structure of D(R) in this case. The paper is organized as follows. ?1 contains a number of basic results about the differential operators on commutative rings. In ?2, the algebraic tool used in proving D(R) is noetherian is developed. This result overlaps with the independent work of J. C. Robson and L. W. Small [11]. ??3 and 4 contain the main results on the structure of D(R) when dim X < 1. This work was motivated by the calculations of I. M. Musson [10]. These results were independently obtained by S. P. Smith and J. T. Stafford [12] in the case that X is an irreducible curve over an algebraically closed field of characteristic zero. ?5 contains an example of a nonreduced k-algebra R with Krull dimension one, such that D(R) is right but not left noetherian. Received by the editors May 22, 1986 and, in revised form, April 13, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A33; Secondary 13B10. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page