Showing papers on "Unit (ring theory) published in 1981"
1 Mar 1981
TL;DR: McDonald et al. as discussed by the authors showed that the primitive condition implies that a polynomial whose values generate the unit ideal actually takes on an invertible value, and showed that such a condition applies to a large class of rings.
Abstract: Let R be a commutative ring. Assume that every polynomial whose values generate the unit ideal actually takes on an invertible value. Then projective R-modules split into cyclic summands, and those of constant rank are free. A ring R (commutative with 1) satisfies the primitive condition if each f(x) = ao + + anXn that is primitive (Y:(aiR) = R) has some b in R with f(b) a unit. This condition, which guarantees the existence of many units in R, was introduced by van der Kallen [13]; he gave examples of rings satisfying the condition and established properties of K2(R) for such R. Subsequently it was shown [4], [5], [6] that the condition implies pleasant structural results about GL2(R) and Aut(GL2(R)). One step in this was a computational argument [6, 11.3] proving that if Q is a rank one direct summand of F where F is free of rank 2, then Q is free. We here will see that a much more general result is true. It actually applies to a slightly larger class of rings, and we begin by discussing them. I. Let f(X1, ... , Xn) be a polynomial over a ring R. We will say that f has local unit values if for each maximal ideal M of R there are b,, . . ., bn in RM with f(bl, ... , bn) invertible in RM. We can here replace the bi by elements of R congruent to them modulo M, so the condition says that not all values of f are in M; in other words, the values of f should generate the unit ideal of R. (This implies, of course, that the coefficients of f must generate the unit ideal.) We say that f has unit values if somef(b1, . . ., bn) is actually invertible in R. The rings we care about will be those in which every f with local unit values has unit values. Since elements are invertible iff they are so modulo the Jacobson radical J(R), it is evident that R has this property iff R/J(R) does. In particular, semilocal rings have the property. It is also easy to see that a product HIRi has the property iff all the factors do (consider maximal ideals of the form Mi x fljH i Rj). Further examples of rings with this property are given by the following propositions. PROPOSITION. Let R be a ring for which R/J(R) is von Neumann regular (= absolutely flat). Then polynomials with local unit values have unit values. PROOF. Replacing R by R/J(R), we may assume it is von Neumann regular. Suppose f has local unit values. For each maximal M pick b = (bl, ... , bn) with Received by the editors August 28, 1980. 1980 Mathematics Subject Classification Primary 13CO5. 'The work of both authors was supported in part by the National Science Foundation. ? 1981 American Mathematical Society 0002-9939/8 1/0000-0503/$02.00 455 This content downloaded from 157.55.39.231 on Thu, 06 Oct 2016 04:38:41 UTC All use subject to http://about.jstor.org/terms 456 B. I{. McDONALD AND W. C. WATEIRHOUSE f(b) a unit at M; then f(b) is still a unit on a neighborhood of M in Spec R. Since Spec R is a Boolean space, we can refine this covering to a finite covering by disjoint clopen sets U where we have f(bu) invertible on U. We may then choose b agreeing with bu on U, and f(b) will be invertible. E] This result could also be deduced from [2, Proposition 2]. The argument shows more generally that if we have a sheaf of rings over a Boolean space and the fibers have our property, so does the ring of global sections. We should also point out that by [1, 11.4, Exercise 16, p. 173] we have the following special case of the proposition: COROLLARY. Let R be zero-dimensional. Then polynomials with local unit values have unit values. LI PROPOSITION. Let S be an R-algebra which is a finitely generated free R-module. Suppose that over R, all polynomials with local unit values have unit values. Then the same is true over S. PROOF. Let s1, ... , sm be a basis of S over R. Given f(X1, ... , XJ) over S, take indeterminates Y 11..., Ymn, and define a polynomial g( Y) over R as the norm (from S to R) of f(> si Yi, . .. , E si Yi). Then setting Xj = E sirij makes f(X) invertible iff g(rij) is invertible, since units and only units have unit norms. Assume now that f has local unit values. If M is any maximal ideal of R, then SM is semilocal, so f has unit values in SM. Hence g has unit values in RM. By hypothesis then g has unit values in R. C] The main theorem will automatically allow us to replace "free" by "projective of constant rank" in this result. We now show exactly how our property is related to the primitive condition mentioned in the introduction. For this we need a pair of simple lemmas. LEMMA. Let R satisfy the primitive condition. Let fi(X) = E aXJ be a finite sequence of polynomials with Eij(aijR) = R. Then there is some b in R with E(fi(b)R) = R. PROOF. Choose an integer m greater than the degrees of all fi, and let g(X) = L f(X)Xm'. All aij occur as coefficients in g, so g is primitive. Hence some g(b) = E f(b)b'm is a unit, and in particular 2(fi(b))R = R. E] This allows us to deduce a multivariable extension of the condition: LEMMA. Let R satisfy the primitive condition. Let f(XI,... , XJ) = E a,Xa be a polynomial with E(a,R) = R. Then there are bl, . .. , b, in R with f(b1, .. ., bn) invertible. PROOF. Rewrite f as E, f,(X1)X8, where ,B = (i2, . .. , in). All the a, appear as the coefficients of the polynomials ff(XI). By the lemma there is some b, such that E(ffi(bj)R) = R. Then f(bl, X2,... Xn) again satisfies the hypothesis of the lemma, and the result follows by induction. El PROPOSITION. A ring R satisfies the primitive condition iff (1) every polynomial with local unit values has unit values and (2) every residue field R/ M is infinite. This content downloaded from 157.55.39.231 on Thu, 06 Oct 2016 04:38:41 UTC All use subject to http://about.jstor.org/terms PROJECTIVE MODULES OVER RINGS 457 PROOF. If R/M has finite cardinality q, then Xq X is a primitive polynomial with all values in M. Thus if R satisfies the primitive condition, (2) must hold. And any f(XI, .. ., X",) with local unit values has coefficients generating the unit ideal, so (1) holds by the last lemma. Conversely, if f is a primitive polynomial, then it is nontrivial modulo M, so by (2) it has a nonzero value modulo M. This means it has local unit values, so by (1) it has unit values. E1 This shows in particular that rings with our property can be very far from dimension zero. Indeed [3], if A is any ring and S is the set of primitive polynomials in A[x], then R = S-'A[x] satisfies the primitive condition and has maximal ideal space identical with that of A.
35 citations
TL;DR: It is shown that no finite union of congruence classes [w], w being an arbitrary element of the free monoid {a, b}∗ with unit 1, is a context-free language if the congruent is defined by the single pair (abbaab, 1).
33 citations
TL;DR: In this paper, it was shown that a cardinal with cf(τ)>ℵ0 has a quotient isomorphic to l∞(κ) if and only if [0, 1]r is a continuous image of the unit ballE′1 ofE′, provided with the w*-topology.
Abstract: Letτ be a cardinal with cf(τ)>ℵ0. Then a Banach spaceE contains a subspace isomorphic toll(τ) if and only if [0,1]r is a continuous image of the unit ballE′1 ofE′, provided with the w*-topology. It follows that, for each cardinalκ, ifE′1 contains a copy ofβκ, thenE has a quotient isomorphic tol∞(κ). In this situation we show thatE has even a quotientisometric tol∞(κ).
27 citations
TL;DR: In this article, it was shown that under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. This result provides a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces?
Abstract: Let Hp(m), 0 < p ^ oo, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of Hp{m), p Φ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of Hp(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly. 1* Introduction* Many results have been obtained concerning isometries of Hardy spaces in the theory of uniform algebras. The most fundamental result in this direction is due to de Leeuw, Rudin, and Wermer [2], which states that an automorphism of the classical Hardy space H°°{T) is induced via composition with the unit circle T of a fractional linear transformation of the unit disc onto itself. Their work was carried on independent of Nagasawa [13], who also described the isometries of ίί oo(Tr) onto itself. On the other hand, Arens [1] completely determined the automorphisms of the uniform algebra of analytic functions on a compact abelian group K whose dual group Γ is archimedean ordered (cf. [11]). This result was extended by Muhly [11] to the uniform algebra of analytic functions induced by a flow which has no periodic orbits. Moreover Muhly [12] has recently given, among other things, the following interesting generalization of this result of Arens to the case of isometries of Hardy spaces Hp(m), p Φ 2, on K: Under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. The purpose of this paper is to remove the assumption on Γ. This result provide a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces
20 citations
TL;DR: In this paper, an eight unit ring expansion method was proposed to proceed by either a [5,5] or consecutive [3,3] sigmatropic shift, respectively.
18 citations
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TL;DR: The unit of the group rings of the dihedral groups D2m (m square free odd) over the rational integer ring are represented by matrices over algebraic number fields as discussed by the authors.
7 citations
TL;DR: In this paper, the minimal length of expressions of an isometry in a symplectic group Sp n (V ) by a product of transvections under the assumption that V is an n -ary nonsingular alternating space over a quasi semilocal semihereditary ring with 2 as a unit was determined.
7 citations
TL;DR: Two new erythrinan alkaloids, erythlaurine and erythramide, were isolated from Cocculus laurifolius DC. and their structures elucidated, featuring a directly attached C1-unit to the aromatic ring at C15 position, expanding the alkaloid constituents of this plant.
Abstract: - Two new erythrinan alkaloids named erythlaurine and erythramide were isolated from the leaves of Cocculus laurifolius DC. (Menispermaceae) and their structures were elucidated by chemical and spectral studies as (1) and (2), respectively. Many erythrinan alkaloids have been isolated from the leaves of Cocculus laurifolius DC. (Menispermaceae).lv2) We have already reported the isolation and structure elucidation of erythraculine, having a directly attached C1-unit to the aromatic ring at C15 position, from the same plant.') In continuation of our investigation of the alkaloid constituents of this plant, we isolated two new erythrinan alkaloids named erythlaurine (1) and erythramide (2). In this communication, we wish to report the structure elucidation of these alkaloids.
7 citations
TL;DR: It is shown that the effect of parasitic bottom plate capacitances can be overcome by using a special type of VIS, and the influence of the top plate parasitics on the filter properties is discussed.
Abstract: Describes a seventh-order unit element switched-capacitor filter based on the voltage invertor switches-switched capacitor (VIS-SC) concept The operation of this filter is described in detail It is shown that the effect of parasitic bottom plate capacitances can be overcome by using a special type of VIS The influence of the top plate parasitics on the filter properties is discussed Experimental results of an integrated NMOS version are given
6 citations
Journal Article•
6 citations
TL;DR: In this article, the authors examined the theory of the special orthogonal group SO (V ) of a symmetric inner product space V over a full ring R with 2 a unit and determined generators, commutator subgroups and the special congruence subgroups.
Journal Article•
Patent•
15 Oct 1981TL;DR: In this article, the authors proposed a feed unit consisting of a feed pump which dips with a tubular intake element into the fluid to be fed and which is provided with a pressure connecting element to which a line which leads to at least one spray jet is connected.
Abstract: A unit is proposed which serves to feed washing fluid from a reservoir to a transparent surface on a motor vehicle, in particular to the cover plate of a motor vehicle light. The feed unit comprises a feed pump which dips with a tubular intake element into the fluid to be fed and which is provided with a pressure connecting element to which a line which leads to at least one spray jet is connected. The free end of the intake element is of pot-shaped construction and has on the floor of the pot at least one orifice for the washing fluid to pass through. The cross-section of the orifice is smaller than the through-flow cross-section of the pressure connecting element enclosed by the pot wall so that extraneous bodies which are located in the reservoir and exceed a specific size cannot pass into the feed pump.
TL;DR: In this article, it was shown that if B is a C ∗ -subalgebra of A containing the unit and such that ξ φ is cyclic in H φ for π φ (B ) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if p φ F is abelian.
TL;DR: In this paper, it was shown that the iterated sum r · m of a stably free module M is free if r is greater than some lower bound, which is the best possible in some cases.
TL;DR: In this paper it was shown that for two types of regular rings A ≥ 3 and n arbitrary, every element of A is a product of an element of (the group of elementary matrices) and a GLr(A) element.
Abstract: In this note we shall prove for two types of regular rings A that every element of is a product of an element of (the group of elementary matrices) and an element of GLr(A), for A ≥3 and n arbitrary. This is a kind of GLr-analogue of results of Lindel and Mohan-Kumar and is an extension of a result of Suslin.
TL;DR: The unit-consumptions are not useful as elements of decision for operators and authorities due to their inaccuracy and the total cost criterion being the main factor for decision-making.
Abstract: It became common to quote energy consumption of transportation operations, generally expressed in g.o.e: by unit of service performed. The author, who does not deny the interest of such figures, to sensitive opinion of the necessary savings, wonders if they are useful as an element of decision. This usefulness seems insignificant to him for the operators. The unit-consumptions do not allow to obtain an accurate compared balance between two real operations. Furthermore, the criterion of the operators is only the total cost. As for the Authorities, they can think of bending theirs choices if the public cost of energy lies above its selling cost. It is the case of the imported energy and particularly of oil. In the field of transportation, this leads to cautious handling of the energetic comparisons between rail and road, for the energy consumed by the first is more and more of national origin.Even rectified about this fact, the unit-consumption cannot be used as elements of comparison between the operations. Each one requires the real of complex calculation of its energetic balance. Consequently, to influence the behaviour through reglementation, i.e. compendiously, does not allow to hope to reach the global economic optimum.The only logical way consists in representing on the cost of imported energy all the expanses it directly or indirectly involves what leads to the increases of certain market prices and sets hard social and political problems.Nevertheless, it is probably such an option which will be chosen in the long run. The seeking of this national price of energy is therefore an impor¬tant asset which should stimulate more research than the perfection of unit-consumption figures without operational interest. Il est devenu d'usage courant de citer des consommations d'énergie d'opérations de transport, généralement exprimées en grammes d'équivalent pétrole par unité de service rendu. L'auteur, qui reconnaît l'intérêt de tels chiffres pour sensibiliser l'opinion aux économies nécessaires, s'interroge sur leur utilité comme éléments de décision. Cette utilité lui paraît insignifiante pour les opérateurs. Les consommations unitaires ne permettent pas d'obtenir un bilan comparé précis entre deux opérations réelles. De plus, le critère des opérateurs est seulement le coût total. La puissance publique, pour sa part, peut songer à infléchir leurs choix si le coût public de l'énergie est supérieur à son coût marchand. C'est le cas pour l'énergie importée, et particulièrement pour le pétrole. En matière de transports, ceci conduit à manier avec précaution les comparaisons énergétiques entre le rail et la route, l'énergie consommée par le premier étant de plus en plus d'origine nationale.Même corrigées de ce fait, les consommations unitaires ne peuvent servir d'éléments de comparaison entre opérations. Chacune exige le calcul réel, et complexe, de son bilan énergétique. Il en résulte qu'agir sur les comportements par voie réglementaire, donc sommaire, ne permet pas d'espérer atteindre l'optimum économique global.La seule voie logique est de représenter sur le coût de l'énergie importée toutes les dépenses qu'elle entraîne directement ou indirectement. Ceci, menant à la majoration de certains prix du marché, pose des problèmes sociaux et politiques difficiles.Néanmoins, c'est probablement vers une telle option qu'on s'achemine à long terme. La recherche de ce prix national de l'énergie est donc un objectif important qui devrait susciter plus de recherches que la perfection de chiffres de consommations unitaires sans intérêt opérationnel.
TL;DR: It is shown that there is essentially only one way of arranging 240 nonoverlapping unit spheres in R 8 (resp.R 24) so that they all touch another unit sphere Ω n, and the following tight spherical t-designs are also unique.
Abstract: We show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R 8 (resp. R 24) so that they all touch another unit sphere Ω n , and only one way of arranging 56 (resp. 4600) spheres in R 8 (resp. R 24) so that they all touch two further, touching spheres. The following tight spherical t-designs are also unique: the 5-design in Ω7, the 7-designs in Ω8 and Ω23 and the 11-design in Ω24.