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Chibudu right hand 1979

Contributions to the theory of group rings

  • Authors: Groenewald, Nicolas Johannes
  • Date: 1979
  • Subjects: Group rings Group theory -- Mathematics
  • Language: English
  • Type: Thesis , Doctoral , PhD
  • Identifier: vital:5391 , http://hdl.handle.net/10962/d1001980
  • Description: Chapter 1 is a short review of the main results in some areas of the theory of group rings. In the first half of Chapter 2 we determine the ideal theoretic structure of the group ring RG where G is the direct product of a finite Abelian group and an ordered group with R a completely primary ring. Our choice of rings and groups entails that the study centres mainly on zero divisor ideals of group rings and hence it contributes in a small way to the zero divisor problem. We show that if R is a completely primary ring, then there exists a one-one correspondence of the prime zero divisor ideals in RG and RG¯, G finite cyclic of order n. If R is a ring with the property α, β € R, then αβ = 0 implies βα = 0, and S is an ordered semigroup, we show that if ∑α¡s¡ ∈ RS is a divisor of zero, then the coefficients α¡ belong to a zero divisor ideal in R. The converse is proved in the case where R is a commutative Noetherian ring. These results are applied to give an account of the zero divisors in the group ring over the direct product of a finite Abelian group and an ordered group with coefficients in a completely primary ring. In the second half of Chapter 2 we determine the units of the group ring RG where R is not necessarily commutative and G is an ordered group. If R is a ring such that if α, β € R and αβ = 0, then βα = 0, and if G is an ordered group, then we show that ∑αg(subscript)g is a unit in RG if and only if there exists ∑βh(subscript)h in RG such that∑αg(subscript)βg(subscript)-1 = 1 and αg(subscriptβh is nilpotent whenever GH≠1. We also show that if R is a ring with no nilpotent elements ≠0 and no idempotents ≠0,1, then RG has only trivial units. In this chapter we also consider strongly prime rings. We prove that RG is strongly prime if R is strongly prime and G is an unique product (u.p.) group. If H ⊲ G such that G/H is right ordered, then it is shown that RG is strongly prime if RH is strongly prime. In Chapter 3 results are derived to indicate the relations between certain classes of ideals in R and RG. If δ is a property of ideals defined for ideals in R and RG, then the "going up" condition holds for δ-ideals if Q being a δ-ideal in R implies that QG is a δ-ideal in RG. The "going down" condition is satisfied if P being a δ-ideal in RG implies that P∩ R is a δ-ideal in R. We proved that the "going up" and "going down" conditions are satisfied for prime ideals, ℓ-prime ideals, q-semiprime ideals and strongly prime ideals. These results are then applied to obtain certain relations between different radicals of the ring R and the group ring (semigroup ring) RG (RS). Similarly, results about the relation between the ideals and the radicals of the group rings RH and RG, where H is a central subgroup of G, are obtained. For the upper nil radical we prove that ⋃(RG) (RH) ⊆ RG, H a central subgroup of G, if G/H is an ordered group . If S is an ordered semigroup, however, then ⋃(RS) ⊆ ⋃(R)S for any ring R. In Chapter 4 we determine relations between various radicals in certain classes of group rings. In Section 4.3, as an extension of a result of Tan, we prove that P(R)G = P(RG) , R a ring with identity , if and only if the order of no finite normal subgroup of G is a zero divisor in R/P(R). If R is any ring with identity and H a normal subgroup of G such that G/H is an ordered group, we show that ⊓(RH)·RG = ⋃(RG) = ⊓(RG) , if ⋃(RH) is nilpotent. Similar results are obtained for the semigroup ring RS, S ordered. It is also shown if R is commutative and G finite of order n, then J(R)G = J(RG) if and only if n is not a zero divisor in R/J(R), J(R) being the Jacobson radical of R. For the Brown HcCoy radical we determine the following: If R is Brown McCoy semisimple or if R is a simple ring with identity, then B(RG) = (0), where G is a finitely generated torsion free Abelian group. In the last section we determine further relations between some of the previously defined radicals, in particular between P(R), U(R) and J(R). Among other results, the following relations between the abovementioned radicals are obtained: U(RS) = U(R)S = P(RS) = J(RS) where R is a left Goldie ring and S an ordered semigroup with unity
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  • Date Issued: 1979

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Creedo from the Missa Zimbabwe

Debut- Hurbet Soudant.

  • Authors: Soudant, Hurbet
  • Date: 1979
  • Subjects: UNCATALOGUED
  • Type: Image
  • Identifier: http://hdl.handle.net/10962/237418 , vital:50513 , DTC-177
  • Description: Clipping of an article about International conductor Hurbet Soudant. , Use of this resource is governed by the terms and conditions of the Creative Commons "Attribution-NonCommercial-ShareAlike" License (http://creativecommons.org/licenses/by-nc-sa/2.0/) (NEVER CAHNGE THIS FIELD)
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  • Date Issued: 1979

Die vooruitsig vir Rhodes Universiteit vir 1979

  • Authors: Henderson, Derek Scott
  • Date: 1979
  • Language: Afrikaans
  • Type: text
  • Identifier: vital:7390 , http://hdl.handle.net/10962/d1017279
  • Description: Rhodes Universiteit gaan nog 'n Nuwe Jaar tegemoet, versterk en bemoedig deur die gebeure en behalings wat agter ons le. Ons het so pas, byvoorbeeld, 'n eeu van tersiere opvoeding in Grahamstad herdenk. Dit was eers in 1904 dat die baanbrekende Kollege-afdeling van St Andrew's plek gemaak het vir die nuwe Rhodes-Universiteitskollege. In September 1979 sal ons die 75ste herdenking van daardie gebeurtenis vier met 'n week van feestelikhede, insluitende 'n spesiale gradeplegtigheid, lesings, vertonings en musiek- en dramauitvoerings.
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  • Date Issued: 1979

Dimbireni Bompa

Dispersal of the cochineal insect Dactylopius Austrinus de Lotto (Homoptera : Dactylopiidae)

  • Authors: Gunn, Brian Howard
  • Date: 1979
  • Subjects: Cochineal insect--South Africa , Cochineal insects , Biological control , Jointed cactus , South Africa
  • Language: English
  • Type: Thesis , Doctoral , PhD
  • Identifier: vital:5596 , http://hdl.handle.net/10962/d1002046
  • Description: Dispersal of the cochineal insect Dactylopius austrinus De Lotto, introduced into South Africa in 1932 as a biological control agent against jointed cactus Opuntia aurantiaca Lindley, was investigated. Zimmermann et al. (l974) suggested that the apparent failure of this insect to control jointed cactus infestations is due to limited dispersal of the first instar nymphs (hereafter referred to as crawlers). Studies on crawler morphology have shown a clear sexual dimorphism in the pattern and development of filaments on the head, thorax and abdomen of male and female crawlers. This enabled differentiation between the sexes with respect to terminal velocities, behaviour and survival of crawlers which have shown that the crawlers, especially the females, are well adapted to dispersal. Long filaments on the head, thorax and abdomen of the female crawlers, that are restricted to dispersal in the crawler stage (as later instars are sessile) and a behaviour directed towards "take-off" enhance the potential for dispersal. The more sedentary males, with long filaments restricted to the abdomen, are able to disperse as winged adults. The principal factors influencing the timing of dispersal and number of crawlers blown from the host plant are wind and temperature; the latter determining the number of crawlers moving on the host plant. Dispersal is confined to the period between 06h00 and 20h00 and it was possible to correlate the pattern of crawler dispersal with wind patterns. The general equation of Taylor (1978) provided an adequate description of horizontal distribution of D.austrinus crawlers in all directions. Wind dispersal of the apterous crawlers is restricted by the low height of jointed cactus plants. Horizontal distribution is limited (generally less than 10 m) although a small proportion of crawlers carried vertically upwards by turbulence or convection currents are sufficiently hardy to survive long range displacement. It is suggested that the small size of the host plant will also reduce effective colonization as the canopy area provides a small target for the wind-blown crawlers. The limitation on dispersal due to the low height of the host plant suggested a system for artificially enhancing crawler dispersal from elevated towers in the field. Evaluation of this system confirmed that it would be practical to augment or introduce cochineal into jointed cactus infestations to enhance the biocontrol potential of this insect. This offers an alternative to chemical control, that has so far failed to control the spread of jointed cactus despite an intensive and expensive herbicide program
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  • Date Issued: 1979

Dudley Tito plays with musicians of Ifindo at the PE Opera house.

  • Authors: Tito, Dudley
  • Date: 1979
  • Subjects: UNCATALOGUED
  • Type: Image
  • Identifier: http://hdl.handle.net/10962/236019 , vital:50368 , DTC-077
  • Description: Clipping of an article on Dudley Tito playing at the Opera House. , Use of this resource is governed by the terms and conditions of the Creative Commons "Attribution-NonCommercial-ShareAlike" License (http://creativecommons.org/licenses/by-nc-sa/2.0/) (NEVER CAHNGE THIS FIELD)
  • Full Text: false
  • Date Issued: 1979

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