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
Some aspects of the construction and implementation of error-correcting linear codes
- Authors: Booth, Geoffrey L
- Date: 1978
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:20967 , http://hdl.handle.net/10962/5718
- Description: From Conclusion: The study of error-correcting codes is now approximately 25 years old. The first known publication on the subject was in 1949 by M. Golay, who later did much research into the subject of perfect codes. It has been recently established that all the perfect codes are known. R.W. Hamming presented his perfect single-error correcting codes in 1950, in ~n article in the Bell System Technical Journal. These codes turned out to be a special case of the powerful Bose-Chaudhuri codes which were discovered around 1960. Various work has been done on the theory of minimal redundancy of codes for a given error-correcting performance, by Plotkin, Gilbert, Varshamov and others, between 1950 and 1960. The binary BCH codes were found to be so close to the theoretical bounds that, to date, no better codes have been discovered. Although the BCH codes are extremely efficient in terms of ratio of information to check digits, they are not easily, decoded with a minimal amount of apparatus. Petersen in 1961 described an algorithm for d e coding BCH codes, but this was cumbersome compared with the majority-logic methods of Massey and others. Thus the search began for codes which are easily decoded with comparatively simple apparatus. The finite geometry codes which were described by Rudolph in a 1964 thesis were examples of codes which are easily decoded 58 by a small number of steps of majority logic. The simplicial codes of Saltzer are even better in this respect, since they can be decoded by a single step of majority logic, but are rather inefficient . The applications of coding theory have changed over the years, as well. The first computers were huge circuits of relays, which were unreliable and prone to errors. Error correcting codes were required to minimise the possibility of incorrect results. As vacuum tubes and later transistorised circuits made computers more reliable, the need for sophisticated and powerful codes in the computer world diminished. Other used presented themselves however, for example the control systems of unmanned space craft. Because of the difficulty of sending and receiving messages in this case, · very powerful codes were required. Other uses were found in transmission lines and telephone exchanges. The codes considered in this dissertation have, for the most part, been block codes for use on the binary symmetric channel. There are, however, several other applications, such as codes for use on an erasure channel, where bits are corrupted so as to be unrecognizable, rather than changed. There are also codes for burst-error correction, where chennel noise is not randomly distributed, but occurs in "bursts" a few bits long. Certain cyclic codes are of application in these cases. The theory of error correcting codes has risen from virtual non-existence in 1950 to a major and sophisticated part of communication theory. Judging from the articles in journals, it promises to be the subject of a great deal of research for some years to come.
- Full Text:
- Date Issued: 1978
- Authors: Booth, Geoffrey L
- Date: 1978
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:20967 , http://hdl.handle.net/10962/5718
- Description: From Conclusion: The study of error-correcting codes is now approximately 25 years old. The first known publication on the subject was in 1949 by M. Golay, who later did much research into the subject of perfect codes. It has been recently established that all the perfect codes are known. R.W. Hamming presented his perfect single-error correcting codes in 1950, in ~n article in the Bell System Technical Journal. These codes turned out to be a special case of the powerful Bose-Chaudhuri codes which were discovered around 1960. Various work has been done on the theory of minimal redundancy of codes for a given error-correcting performance, by Plotkin, Gilbert, Varshamov and others, between 1950 and 1960. The binary BCH codes were found to be so close to the theoretical bounds that, to date, no better codes have been discovered. Although the BCH codes are extremely efficient in terms of ratio of information to check digits, they are not easily, decoded with a minimal amount of apparatus. Petersen in 1961 described an algorithm for d e coding BCH codes, but this was cumbersome compared with the majority-logic methods of Massey and others. Thus the search began for codes which are easily decoded with comparatively simple apparatus. The finite geometry codes which were described by Rudolph in a 1964 thesis were examples of codes which are easily decoded 58 by a small number of steps of majority logic. The simplicial codes of Saltzer are even better in this respect, since they can be decoded by a single step of majority logic, but are rather inefficient . The applications of coding theory have changed over the years, as well. The first computers were huge circuits of relays, which were unreliable and prone to errors. Error correcting codes were required to minimise the possibility of incorrect results. As vacuum tubes and later transistorised circuits made computers more reliable, the need for sophisticated and powerful codes in the computer world diminished. Other used presented themselves however, for example the control systems of unmanned space craft. Because of the difficulty of sending and receiving messages in this case, · very powerful codes were required. Other uses were found in transmission lines and telephone exchanges. The codes considered in this dissertation have, for the most part, been block codes for use on the binary symmetric channel. There are, however, several other applications, such as codes for use on an erasure channel, where bits are corrupted so as to be unrecognizable, rather than changed. There are also codes for burst-error correction, where chennel noise is not randomly distributed, but occurs in "bursts" a few bits long. Certain cyclic codes are of application in these cases. The theory of error correcting codes has risen from virtual non-existence in 1950 to a major and sophisticated part of communication theory. Judging from the articles in journals, it promises to be the subject of a great deal of research for some years to come.
- Full Text:
- Date Issued: 1978
Some aspects of the theory, application, and computation of generalised inverses of matrices
- Authors: Cretchley, Partricia C
- Date: 1977
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/6212 , vital:21063
- Description: The idea of generalising the classical notion of the inverse of a non-singular matrix arose as far back as in 1920, but it was not until the late fifties that the development of the theory gained any impetus. Since then , as is the case in the development of many new concepts , work done in parallel in various parts of the world has resulted in a great deal of untidiness in the literature : confusion over terminology , and even duplication of theory. More recently, however, some attempts have been made to bring together people active in the field of generalised inverses, in order to reach consensus on some aspects of definition and terminology, and to publish more general works on the subject. Towards this purpose, a symposium on the theory and application of generalised inverses of matrices was held in Lubbock, Texas, and its proceedings published in 1968 (see [25] ). A few other works of this nature (see [4], (19a] ) have appeared , but the bulk of the literature still comprises numerous diverse papers offering further ideas on the theoretical properties which these matrices have , and drawing attention to their application in areas of statistics , numerical analysis , filtering , modern control and estimation theory, pattern recognition and many others. This essay offers a look at generalised inverses in the following way: firstly a broad basis and background is established in the first three chapters to provide greater understanding of the motivation for the remaining chapters, where the approach then changes to become far more detailed. Within this general framework, Chapter 1 offers a brief glimpse of the history and development of work in the field. In Chapter 2 some of the most significant properties of these inverses are described, while in Chapters 3 and 4 and 5 attention is given to interesting and remarkable computational algorithms relating to generalised inverses (some well suited to machine processing). The material of Chapters 4 and 5 is largely due to Decell, Stallings and Boullion, and Tanabe, in [6], [24] and [27], respectively, while the source of material for the first three chapters is the literature generally, with Penrose's two papers providing a rough framework for Chapters 1 and 2 (see [17]).
- Full Text:
- Date Issued: 1977
- Authors: Cretchley, Partricia C
- Date: 1977
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/6212 , vital:21063
- Description: The idea of generalising the classical notion of the inverse of a non-singular matrix arose as far back as in 1920, but it was not until the late fifties that the development of the theory gained any impetus. Since then , as is the case in the development of many new concepts , work done in parallel in various parts of the world has resulted in a great deal of untidiness in the literature : confusion over terminology , and even duplication of theory. More recently, however, some attempts have been made to bring together people active in the field of generalised inverses, in order to reach consensus on some aspects of definition and terminology, and to publish more general works on the subject. Towards this purpose, a symposium on the theory and application of generalised inverses of matrices was held in Lubbock, Texas, and its proceedings published in 1968 (see [25] ). A few other works of this nature (see [4], (19a] ) have appeared , but the bulk of the literature still comprises numerous diverse papers offering further ideas on the theoretical properties which these matrices have , and drawing attention to their application in areas of statistics , numerical analysis , filtering , modern control and estimation theory, pattern recognition and many others. This essay offers a look at generalised inverses in the following way: firstly a broad basis and background is established in the first three chapters to provide greater understanding of the motivation for the remaining chapters, where the approach then changes to become far more detailed. Within this general framework, Chapter 1 offers a brief glimpse of the history and development of work in the field. In Chapter 2 some of the most significant properties of these inverses are described, while in Chapters 3 and 4 and 5 attention is given to interesting and remarkable computational algorithms relating to generalised inverses (some well suited to machine processing). The material of Chapters 4 and 5 is largely due to Decell, Stallings and Boullion, and Tanabe, in [6], [24] and [27], respectively, while the source of material for the first three chapters is the literature generally, with Penrose's two papers providing a rough framework for Chapters 1 and 2 (see [17]).
- Full Text:
- Date Issued: 1977
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
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