Showing papers on "Ring (mathematics) published in 1975"

The Upper Bound Conjecture and Cohen‐Macaulay Rings

Abstract: Let Δ be a triangulation of a (d − 1)-dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i-dimensional faces of Δ is less than or equal to a certain explicit number ci(n, d). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen-Macaulay ring.

The predual theorem to the Jacobson-Bourbaki theorem

Abstract: Suppose R S is a map of rings. S need not be an R algebra since R may not be commutative. Even if R is commutative it may not have central image in S. Nevertheless the ring structure on S can be expressed in terms of two maps (S1 a S p s1s2) 'O ~ ~ ~ , R-!eS, which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an R-coring. Suppose R is an overing of B. Let CB = R OB R. There are maps (r, 3 r2 -r, ? 1 0 r2) CB= R OB R >R IB R %B R = (CB) GR (CB), (r, G r2 r,r2) CB=R BR R R. These maps give CB an R-coring structure. The dual *CB is naturally isomorphic to the ring EndBER of B-linear endomorphisms of R considered as a left B-module. In case B happens to be the subring of R generated by 1, we write Cz. Then *Cz is EndzR, the endomorphism ring of R considered as an additive group. This gives a clue how certain R-corings correspond to subrings of R and subrings of EndZR, both major ingredients of the Jacobson-Bourbaki theorem. 1 3 1 is a "grouplike" element in the R-coring CZ (and should be thought of as a generic automorphism of R). Suppose R is a division ring and B a subring which is a division ring. The natural map CZ -CB is a surjective coring map. Conversely if CZ D is a (surjective) coring map then 7r(1 0 1) is a grouplike in D and {r E R I r7r(1 X 1) = 7r(1 3 l)r} is a subring of R which is a division ring. This gives a bijective correspondence between the quotient corings of C ez C and the subrings of R which are division rings. We show how the Jacobson-Bourbaki correspondence is dual to the above correspondence. Received by the editors December 17, 1974. AMS (MOS) subject classifications (1970). Primary 16A74; Secondary 16A40, 16A56.

Jordan derivations on rings

Abstract: I. N. Herstein has shown that-every Jordan derivation on a prime ring not of characteristic 2 is a derivation. This result is extended in this paper to the case of any ring in which 2x 0 O implies x = 0 and which is semiprime or which has a commutator which is not a zero divisor.

The indecomposable representations of the dihedral 2-groups

Abstract: Let K be a field. We will give a complete list of the normal forms of pairs a, b of endomorphisms of a K-vector space such that a 2 b 2 = 0. Thus, we determine the modules over the ring R = K ( X , Y)/(X 2, y2) which are finite dimensional as K-vector spaces; here (X 2, y2) stands for the ideal generated by X 2 and y2 in the free associative algebra K (X, Y) in the variables X and Y. If G is the dihedral group of order 4q (where q is a power of 2) generated by the involutions 91 and 92, and if the characteristic of K is 2, then the group algebra K G is a factor ring of R, and the K G-modules KGM which have no non-zero projective submodule correspond to the K-vector spaces (take the underlying space of ~ M ) together with two endomorphisms a and b (namely multiplication by g ~ 1 and g 2 1 , respectively) such that, in addition to a Z b 2 -0 , also (ab) q = (ba) q = 0 is satisfied. We use the methods of Gelfand and Ponomarev developped in their joint paper on the representations of the Lorentz group, where they classify pairs of endomorphisms a, b such that ab = ba = O. The presentation given here follows closely the functorial interpretation of the Gelfand-Ponomarev result by Gabriel, which he exposed in a seminar at Bonn, and the author would like to thank him for many helpful conversations.

Induction and structure theorems for orthogonal representations of finite groups

Abstract: The equivariant Witt ring GW(, R) of a finite group w over a Dedekind domain R is studied. It is shown that-modulo the prime 2-GW(r, Z) equals the character ring of real representations of w and GW(w, R) equals GW(w, Z) ( W(R). From this, induction theorems a la E. Artin and R. Brauer are derived for GW(-, R) and it is shown how these can be applied towards the computation of L-groups.

Hilbert’s tenth problem for quadratic rings

Abstract: Let A(D) be any quadratic ring; in this paper we prove that Hilbert's tenth problem for A(D) is unsolvable, and we determine which relations are diophantine over A(D).

Quasi-Injective and Pseudo-Infective Modules

Abstract: Let R be a ring with identity not equal to zero. A right R-module is said to be quasi-injective (pseudo-injective) if for every submodule N of M, every R-homomorphism (R-monomorphism) of N into M can be extended to an R-endomorphism of M [7] ([13]).

An implementation of hyper-resolution

Abstract: In this paper the data structures and algorithms which were used to implement hyper-resolution are presented. The algorithms, which do not generate hyper-resolvents by creating sequences of P 1-resolvents, have been used to obtain proofs of THEOREM 1. Let G be a group such that x 3 = e for all x∈G. If h is defined as h(x, y) = xyx′y′ forx, y∈G, then for all x, y∈G, h(h(x, y), y) = e (the identity). THEOREM 2. Let R be a ring such that x 2 = x for all x∈R. Then R is commutative. THEOREM 3. Every subgroup of index 2 is normal. The data structures have been designed so that only a single copy of any literal or term is retained, no matter how often it occurs in the clauses kept. The main advantage of this approach is not the resulting savings in storage, but instead the fact that simultaneously matching a set of literals generates an entire set of hyper-resolvents. A method of extracting a set of “candidates for unification with a given literal” from the data structures is also presented. The result of using this method is a substantial reduction in the number of times a complete unification of two literals must be attempted. The initial results obtained from the program suggest that many resolution algorithms besides hyper-resolution could be enhanced by the use of similar data structures and algorithms.

The Complexity of the Quaternion Product

Abstract: Let X and Y be two quaternions over an arbitrary ring. Eight multiplications are necessary and sufficient for computing the product XY. If the ring is assumed to be commutative, at least seven multiplications are still necessary and eight are sufficient.

On the injectivity and flatness of certain cyclic modules

Abstract: The question of when certain cyclic flat modules of a ring are injective (and vice versa) is studied The consequences of the conditions 'flat' and 'injective' on the simple modules of a ring are discussed Introduction The object of this paper is to consider the relationships between the injectivity and flatness of cyclic modules of the form R/A where A is a two-sided ideal of the ring R In general neither the implication R/A is right injective R/A is right flat nor its reverse need be true However, Ware [11] has recently shown that if R is commutative and A is a maximal ideal, then both implications are true We generalize this observation to noncommutative rings and indicate some conditions under which the injectivity of R/A implies its flatness These considerations lead to Baer's notion of pR-completeness and it turns out that, for simple modules over commutative rings, pR-completeness is equivalent to injectivity, thus yielding a new characterzation of commutative von Neumann regular rings The paper ends with a discussion of pR-completeness and flatness of simple modules over noncommutative rings Terminology The terms ring, module and homomorphism will mean ring with unity, unitary right module and right module homomorphism respectively 'Ideal will mean two-sided ideal 'If R is a ring and M is a right R-module, M will be said to be I-complete for a right ideal I of R if any homomorphism I -* M can be extended to a homomorphism R -f M 1 Simple modules Let A be an ideal of R 'In this section the connection between the injectivity and flatness of the cyclic module R/A is discussed for the case when A is a maximal right ideal Received by the editors October 3, 1973 and, in revised form, January 16, 1974 AMS (MOS) subject classifications (1970) Primary 16A30, 16A50, 16A52

Pseudo-injective modules which are not quasi-injective

Abstract: For certain rings with infinitely many nonisomorphic simple left modules, a method is given for constructing pseudo-injective modules which are not quasi-injective. This method is used to produce examples of such modules over a commutative ring. Let R be a ring with unity. All modules considered here will be unital left R-modules. A module M is called quasi-injective (pseudo-injective) if, for every submodule N of M, every R-homomorphism (R-monomorphism) from N to M can be extended to an R-endomorphism of M [5] ([6]). Every quasiinjective module is pseudo-injective. In previous papers (e.g. [4], [6], [7]), most of the results on pseudo-injective modules are of the form, "if R satisfies a suitable hypothesis, then certain pseudo-injective modules are quasiinjective." The intent of most of the work, then, was to show that pseudoinjectives were generally always quasi-injective (e.g. see the comment at the end of the Introduction to [7]). Indeed, the only two examples of pseudoinjective modules which are not quasi-injective have recently appeared in the literature (see [2] and [4]). Both of these modules have precisely five submodules and have Loewy length 2. In this note, we give a construction for forming pseudo-injective modules which are not quasi-injective. This construction yields examples which answer in the negative the following two questions of S. K. Jain [3] (see also [4]): (i) Is every pseudo-injective module over a commutative ring quasiinjective? (ii) Is every nonsingular pseudo-injective module quasi-injective? Using an example of Fuchs [1], we can also apply our construction to show that a pseudo-injective module which is not quasi-injective may have arbitrarily large Loewy length. Received by the editors February 8, 1974. AMS (MOS) subject classifications (1970). Primary 16A52; Secondary 13C10.

Universal extension for outlet boxes

Abstract: A universal extension ring for electrical outlet boxes is described. It is adapted to be mounted on the ordinary single or two-device outlet box to increase its volume and also to enable single or double switches or receptacles to be secured thereto. The extension ring features removable back panels to accomplish the above objectives.

Note on rings of finite representation type and decompositions of modules

Abstract: Tachikawa has shown that if a ring A is of finite representation type, then each of its left and right modules has a decomposition that complements direct summands. We show that the converse is also true. Anderson and Fuller [1 I posed the problem of determining over which rings does every module have a decomposition M = DAMa that complements direct summands in the sense that whenever K is a direct summand of M, M = K (E (E)3B M) for some B C A. In response, Tachikawa [6] has proved that the modules over a ring of finite representation type have such decompositions. We recall that an artin ring is of finite representation type if it has only a finite number of finitely generated indecomposable left modules. The purpose of this note is to use the results of [1]-[ 5] to show that the converse of Tachikawa's result is also true. Auslander [3] says that a family of R-homomorphisms is noetherian if given a sequence fo fl M -1-* M M .* m0 1 2 in the family, with /i . fl 0 for all i, there is an integer n such that fk is an isomorphism for all k > n, and that the family is conoetherian in case given any sequence

Global dimension of differential operator rings. II

Abstract: This paper is concerned with finding the global homological dimension of the ring of differential operators R[0] over a differential ring R with a single derivation. Examples are constructed to show that R[O] may have finite dimension even when R has infinite dimension. For a commutative noetherian differential algebra R over the rationals, with finite global dimension n, it is shown that the global dimension of R[0] is the supremum of n and one plus the projective dimensions of the modules R/P, where P ranges over all prime differential ideals of R. One application derives the global dimension of the Weyl algebra over a commutative noetherian ring S of finite global dimension, where S either is an algebra over the rationals or else has positive characteristic.

The analytic theory of algebraic numbers

Abstract: 1. The basics of algebraic number theory. An algebraic number field is a field K = Q(a) where a is a zero of an irreducible (over Q) polynomial f(x) with integral coefficients. The degree of K, which we denote by n = n(K) = [K:Q], is the degree of ƒ(%). We write the roots of /(x) = 0 as a, a, • • • ,a ( n ) m such a way that for l^j^r1 = r1(K), a Q) is real, while for j>ru a 0 ) is complex. If we let n = ri+2r2, then it is customary to order the rr=r2(K) complex conjugate pairs of roots so that for r i+ l ^ j ^ r i+ r 2 , a=a\ The a ( , ) s are called the conjugates of a and the fields K ( , =Q(a) are called the conjugate fields of K. If r2 = 0, we say K is totally real and if ri=0, we say K is totally complex. The integers of K are those elements of K which are zeros of a polynomial with integer coefficients and leading coefficient 1. The integers of K form a ring which we denote by o. As is well known, factorization of the integers of K into prime integers is not necessarily unique. Various equivalent ways of remedying this have been used; we follow Dedekind's method. If a i , • • • , ak are elements of X, the set

$\aleph_0$-Categorical Modules

Abstract: It is shown that the first-order theory Th R ( A ) of a countable module over an arbitrary countable ring R is ℵ 0 -categorical if and only if A i finite, n ∈ ω , κ i ≤ ω . Furthermore, Th R (A) is ℵ 0 -categorical for all R -modules A if and only if R is finite and there exist only finitely many isomorphism classes of indecomposable R -modules.

The regular ring and the maximal ring of quotients of a finite Baer *-ring

Abstract: Necessary and sufficient conditions are obtained for extending the involution of a Baer »-ring to its maximal ring of quotients. Berberian's construction of the regular ring of a Baer «-ring is generalized and this ring is identified (under suitable hypotheses) with the maximal ring of quotients.

Sur un Théorème de Shamir

Abstract: en Shamir has proved that any algebraic power series over a free monoid X * , with coefficients in an abelian semiring A , may be calculated by a representation of X * into the A -ring A 〈 H 〉 of a half-free-group H . The same result has been proved by Nivat, with the A -ring A 〈 G 〉 of a free group G . We state here the converse of this result: Any power series defined by such a representation is algebraic. In order to prove that, we use the following material: a particular class of transductions, which we call “regulated rational transductions”. They seem to be the good generalization for power series of the “faithfully rational transductions” for languages.

On the Global Dimensions of D+M

Abstract: This note answers affirmatively a question of the author [4, p. 456], by producing an example of an integrally closed quasi-local non valuation domain of global dimension 3, each of whose overrings is a goingdown ring. Although [4, Proposition 4.5] shows that such an example cannot be constructed by means of restrained power series, an approach via the more general D+M construction succeeds.

Generalized uniserial rings

Abstract: The following conditions are shown to be equivalent: 1) ring A is generalized uniserial (not necessarily artinian); 2) every finitely presented A module is semiserial; 3) A is semiperfect and the projective cover of every indecomposable finitely presented module is indecomposable.

Coordinate theorems for affine Hjelmslev planes

Abstract: It is shown that an affine Hjelmslev plane ℋ is a translation plane if and only if each of its coordinate biternary rings B=〈k, T, T0, 0, 1〉 are linear. Addition and multiplication in the ternary ring 〈k, T, 0, 1〉 are defined by a+b=T(a, 1, b) and a·b= =T(a, b, 0), respectively, and it is proved that every biternary ring of a translation plane has the additional properties that 〈k,+〉 is an abelian group 〈k, +, ·〉 is right distributive, and T(a, 1, b)=T0(a, 1, b). Moreover, if a single linear biternary ring of ℋ has these three properties, then ℋ is a translation plane. It is shown that a translation plane is Desarguesian if and only if it has a linear biternary ring such that T=T0 and 〈k, +, ·〉 is an affine Hjelmslev ring. Hessenberg’s theorem for affine Hjelmslev planes is proved, and a special configurational condition which is equivalent to the commutativity of multiplication in each biternary ring is introduced.