Contributions to the theory of group rings
- Groenewald, Nicolas Johannes
- 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
- Full Text:
- Date Issued: 1979
- 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
- Full Text:
- Date Issued: 1979
Lesniewski's logic aspects of his protothetic, ontology and mereology
- Authors: Norman, Max
- Date: 1979
- Language: English
- Type: Thesis , Masters , MA
- Identifier: vital:21148 , http://hdl.handle.net/10962/6596
- Description: Stanislaw Lesniewski (1886-1939) was professor of Philosophy of Mathematics at the University of Warsaw from 1919 until his death. He played a leading role in the Warsaw school of logic and had a lasting influence on many of its members. Lesniewski constructed his first description of mereology in colloquial language and in the absence of a secure logical foundation. In order to effectively distinguish between the collective and distributive notions of class, further description of the distributive notion was necessary. He therefore formalized the distributive concepts in his theory of ontology. Henceforth "ontology" will be used specifically to refer to this theory of Lesniewski. Finally, the construction of protothetic (a system of propositional logic) provided a sound logical foundation of Lesniewski's ontology and mereology. Protothetic, together with his prescribed rules of procedure and his grammar of semantic categories, also facilitated the formalization of his systems in a logically rigorous manner. All of the researchers acknowledge that one of Lesniewski's most fundamental achievements was the development of his deductive systems (protothetic, ontology and mereology).
- Full Text:
- Date Issued: 1979
- Authors: Norman, Max
- Date: 1979
- Language: English
- Type: Thesis , Masters , MA
- Identifier: vital:21148 , http://hdl.handle.net/10962/6596
- Description: Stanislaw Lesniewski (1886-1939) was professor of Philosophy of Mathematics at the University of Warsaw from 1919 until his death. He played a leading role in the Warsaw school of logic and had a lasting influence on many of its members. Lesniewski constructed his first description of mereology in colloquial language and in the absence of a secure logical foundation. In order to effectively distinguish between the collective and distributive notions of class, further description of the distributive notion was necessary. He therefore formalized the distributive concepts in his theory of ontology. Henceforth "ontology" will be used specifically to refer to this theory of Lesniewski. Finally, the construction of protothetic (a system of propositional logic) provided a sound logical foundation of Lesniewski's ontology and mereology. Protothetic, together with his prescribed rules of procedure and his grammar of semantic categories, also facilitated the formalization of his systems in a logically rigorous manner. All of the researchers acknowledge that one of Lesniewski's most fundamental achievements was the development of his deductive systems (protothetic, ontology and mereology).
- Full Text:
- Date Issued: 1979
Instability in the magnetotail
- Authors: English, Daniel Rowe
- Date: 1977
- Subjects: Magnetotails
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5425 , http://hdl.handle.net/10962/d1011764 , Magnetotails
- Description: The magnetic induction field due to the Earth only would, if undisturbed by any outside agency, resemble macroscopically the field due to a magnetic dipole. Hcwever the field is disturbed by the interplanetary magnetic field, of which the most important component is that of the Sun. If the Sun's magnetic field were effectively steady, it would also be a dipole field, and approximately constant in the region within about twenty earth radii from the earth. Also, if we treat the Sun as a dipole, its dipole axis is roughly normal to the ecliptic plane. The Earth, treated as a dipole, has an axis which is inclined to the normal to the ecliptic plane at an angle which varies daily from a few degrees to nearly a third of a right angle. However, in this paper, it is proposed to treat both dipole axes as contra-parallel and effectively normal to the ecliptic plane, so that a general idea of the combined field can be obtained. Then the effect of a steady field due to the Sun, on the Earth's field would be the formation of a "neutral ring" surrounding the Earth; that is, a closed "neutral line", this being a line of points at each of which the net nagnetic induction is zero. As the point of observation passes through this line, the field changes direction. Intro. p. v.
- Full Text:
- Date Issued: 1977
- Authors: English, Daniel Rowe
- Date: 1977
- Subjects: Magnetotails
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5425 , http://hdl.handle.net/10962/d1011764 , Magnetotails
- Description: The magnetic induction field due to the Earth only would, if undisturbed by any outside agency, resemble macroscopically the field due to a magnetic dipole. Hcwever the field is disturbed by the interplanetary magnetic field, of which the most important component is that of the Sun. If the Sun's magnetic field were effectively steady, it would also be a dipole field, and approximately constant in the region within about twenty earth radii from the earth. Also, if we treat the Sun as a dipole, its dipole axis is roughly normal to the ecliptic plane. The Earth, treated as a dipole, has an axis which is inclined to the normal to the ecliptic plane at an angle which varies daily from a few degrees to nearly a third of a right angle. However, in this paper, it is proposed to treat both dipole axes as contra-parallel and effectively normal to the ecliptic plane, so that a general idea of the combined field can be obtained. Then the effect of a steady field due to the Sun, on the Earth's field would be the formation of a "neutral ring" surrounding the Earth; that is, a closed "neutral line", this being a line of points at each of which the net nagnetic induction is zero. As the point of observation passes through this line, the field changes direction. Intro. p. v.
- Full Text:
- Date Issued: 1977
A computer analysis of some of the Harrison metrics
- Authors: Sadler, Christopher John
- Date: 1975
- Subjects: Computer science -- Mathematics , Software measurement
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5427 , http://hdl.handle.net/10962/d1013151
- Description: In his paper B.K.Harrison concludes with the observation that his "solutions ... are presented as raw material for further research in General Relativity". In the same spirit, the present work started out as an attempt to process that raw material in a production-line powered by a computer. Harrison's solutions uould be fed in at one end, and the finished product, as yet undecided, would appear at the other. In the event, however, the project became more like an exercise in quality control, to continue the analogy. A search was made for algebraic criteria which would distinguish between those solutions which were acceptable for further analysis with particular regard to Gravitational radiation, and those which were not. Regrettably, no criteria could be found which characterised radiative solutions unequivocally, and, at the same time, lent themselves to a computer approach. The result is that the discussion of radiative solutions has had to be relegated to an appendix (Appendix 1), while the main body of the work is concerned with the determination of those quantities (the Newman-Penrose scalars) which would seem to be the foundation of any future computer-based analysis of gravitational radiation. Chapter 1 is an account of the underlying mathematical formulation, defining the terms, concepts and processes involved. In Chapter 2 the transformation of some of the ideas of Chapter 1 into computer software is presented. Chapter 3 is concerned with the specific metrics (the Harrison metrics) and the extent to which they have heen processed. The project has leaned heavily on papers by Harrison for the "raw material", by D' Inverno and Russell Clark, who pioneered the techniques and classified the Harrison metrics, and by Sachs for the treatment of gravitational radiation. However, the analysis of diagonal metrics, the special tetrad of Chapter 2 and the results in Appendix 2 are new.
- Full Text:
- Date Issued: 1975
- Authors: Sadler, Christopher John
- Date: 1975
- Subjects: Computer science -- Mathematics , Software measurement
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5427 , http://hdl.handle.net/10962/d1013151
- Description: In his paper B.K.Harrison concludes with the observation that his "solutions ... are presented as raw material for further research in General Relativity". In the same spirit, the present work started out as an attempt to process that raw material in a production-line powered by a computer. Harrison's solutions uould be fed in at one end, and the finished product, as yet undecided, would appear at the other. In the event, however, the project became more like an exercise in quality control, to continue the analogy. A search was made for algebraic criteria which would distinguish between those solutions which were acceptable for further analysis with particular regard to Gravitational radiation, and those which were not. Regrettably, no criteria could be found which characterised radiative solutions unequivocally, and, at the same time, lent themselves to a computer approach. The result is that the discussion of radiative solutions has had to be relegated to an appendix (Appendix 1), while the main body of the work is concerned with the determination of those quantities (the Newman-Penrose scalars) which would seem to be the foundation of any future computer-based analysis of gravitational radiation. Chapter 1 is an account of the underlying mathematical formulation, defining the terms, concepts and processes involved. In Chapter 2 the transformation of some of the ideas of Chapter 1 into computer software is presented. Chapter 3 is concerned with the specific metrics (the Harrison metrics) and the extent to which they have heen processed. The project has leaned heavily on papers by Harrison for the "raw material", by D' Inverno and Russell Clark, who pioneered the techniques and classified the Harrison metrics, and by Sachs for the treatment of gravitational radiation. However, the analysis of diagonal metrics, the special tetrad of Chapter 2 and the results in Appendix 2 are new.
- Full Text:
- Date Issued: 1975
Remarks on formalized arithmetic and subsystems thereof
- Brink, C
- Authors: Brink, C
- Date: 1975
- Subjects: Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5424 , http://hdl.handle.net/10962/d1009752 , Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory
- Description: In a famous paper of 1931, Gödel proved that any formalization of elementary Arithmetic is incomplete, in the sense that it contains statements which are neither provable nor disprovable. Some two years before this, Presburger proved that a mutilated system of Arithmetic, employing only addition but not multiplication, is complete. This essay is partly an exposition of a system such as Presburger's, and partly an attempt to gain insight into the source of the incompleteness of Arithmetic, by linking Presburger's result with Gödel's.
- Full Text:
- Date Issued: 1975
- Authors: Brink, C
- Date: 1975
- Subjects: Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5424 , http://hdl.handle.net/10962/d1009752 , Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory
- Description: In a famous paper of 1931, Gödel proved that any formalization of elementary Arithmetic is incomplete, in the sense that it contains statements which are neither provable nor disprovable. Some two years before this, Presburger proved that a mutilated system of Arithmetic, employing only addition but not multiplication, is complete. This essay is partly an exposition of a system such as Presburger's, and partly an attempt to gain insight into the source of the incompleteness of Arithmetic, by linking Presburger's result with Gödel's.
- Full Text:
- Date Issued: 1975
Twistors in curved space
- Ward, R S (Richard Samuel), 1951-
- Authors: Ward, R S (Richard Samuel), 1951-
- Date: 1975
- Subjects: Twistor theory , Space and time
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5429 , http://hdl.handle.net/10962/d1013472
- Description: From the Introduction, p. 1. During the past decade, the theory of twistors has been introduced and developed, primarily by Professor Roger Penrose, as part of a long-term program aimed at resolving certain difficulties in present-day physical theory. These difficulties include, firstly, the problem of combining quantum mechanics and general relativity, and, secondly, the question of whether the concept of a continuum is at all relevant to physics. Most models of space-time used in general relativity employ the idea of a manifold consisting of a continuum of points. This feature of the models has often been criticised, on the grounds that physical observations are essentially discrete in nature; for reasons that are mathematical, rather than physical, the gaps between these observations are filled in a continuous fashion (see, for example, Schrodinger (I), pp.26-31). Although analysis (in its generally accepted form) demands that quantities should take on a continuous range of values, physics, as such,does not make such a demand. The situation in quantum mechanics is not all that much better since, although some quantities such as angular momentum can only take on certain discrete values, one still has to deal with the complex continuum of probability amplitudes. From this point of view it would be desirable to have all physical laws expressed in terms of combinatorial mathematics, rather than in terms of (standard) analysis.
- Full Text:
- Date Issued: 1975
- Authors: Ward, R S (Richard Samuel), 1951-
- Date: 1975
- Subjects: Twistor theory , Space and time
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5429 , http://hdl.handle.net/10962/d1013472
- Description: From the Introduction, p. 1. During the past decade, the theory of twistors has been introduced and developed, primarily by Professor Roger Penrose, as part of a long-term program aimed at resolving certain difficulties in present-day physical theory. These difficulties include, firstly, the problem of combining quantum mechanics and general relativity, and, secondly, the question of whether the concept of a continuum is at all relevant to physics. Most models of space-time used in general relativity employ the idea of a manifold consisting of a continuum of points. This feature of the models has often been criticised, on the grounds that physical observations are essentially discrete in nature; for reasons that are mathematical, rather than physical, the gaps between these observations are filled in a continuous fashion (see, for example, Schrodinger (I), pp.26-31). Although analysis (in its generally accepted form) demands that quantities should take on a continuous range of values, physics, as such,does not make such a demand. The situation in quantum mechanics is not all that much better since, although some quantities such as angular momentum can only take on certain discrete values, one still has to deal with the complex continuum of probability amplitudes. From this point of view it would be desirable to have all physical laws expressed in terms of combinatorial mathematics, rather than in terms of (standard) analysis.
- Full Text:
- Date Issued: 1975