M.Sc. / One of the earliest results (1955) in the theory of derivations is the celebrated theorem of I. M. Singer and J. Wermer [14] which asserts that every bounded derivation on a commutative Banach algebra has range contained in the radical. However, they immediately conjectured that their result will still hold if the boundedness condition was dropped. This conjecture of Singer and Wermer was confirmed only in 1988, by M. P. Thomas [23], when he showed that every derivation (bounded or unbounded) on a commutative Banach algebra has range contained in the radical. But it is not known whether an analogue of the Kleinecke-Shirokov Theorem holds for everywhere defined unbounded derivation.

M.Sc. (Mathematics) / Please refer to full text to view abstract

M.Sc. (Mathematics) / Please refer to full text to view abstract

M.Ed. (Subject Didactics) / Please refer to full text to view abstract

This thesis is aimed at generalizing notions of rings to modules. In par-ticular, notions of completely prime ideals, s-prime ideals, 2-primal rings and nilpotency of elements of rings are respectively generalized to completely prime submodules and classical completely prime submodules, s-prime submodules, 2-primal modules and nilpotency of elements of modules. Properties and rad-icals that arise from each of these notions are studied.

by Au Yun-Nam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 133-137). / Introduction --- p.iv / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Some Semigroup Properties --- p.1 / Chapter 1.2 --- General Properties of Semigroup Algebras --- p.5 / Chapter 1.3 --- Group Algebras --- p.7 / Chapter 1.3.1 --- Some Basic Properties of Groups --- p.7 / Chapter 1.3.2 --- General Properties of Group Algebras --- p.8 / Chapter 1.3.3 --- Δ-Method for Group Algebras --- p.10 / Chapter 1.4 --- Graded Algebras --- p.12 / Chapter 1.5 --- Crossed Products and Smash Products --- p.14 / Chapter 2 --- Radicals of Graded Rings --- p.17 / Chapter 2.1 --- Jacobson Radical of Crossed Products --- p.17 / Chapter 2.2 --- Graded Radicals and Reflected Radicals --- p.18 / Chapter 2.3 --- Radicals of Group-graded Rings --- p.24 / Chapter 2.4 --- Algebras Graded by Semilattices --- p.26 / Chapter 2.5 --- Algebras Graded by Bands --- p.27 / Chapter 2.5.1 --- Hereditary Radicals of Band-graded Rings --- p.27 / Chapter 2.5.2 --- Special Band-graded Rings --- p.30 / Chapter 3 --- Radicals of Semigroup Algebras --- p.34 / Chapter 3.1 --- Radicals of Polynomial Rings --- p.34 / Chapter 3.2 --- Radicals of Commutative Semigroup Algebras --- p.36 / Chapter 3.2.1 --- Commutative Cancellative Semigroups --- p.37 / Chapter 3.2.2 --- General Commutative Semigroups --- p.39 / Chapter 3.2.3 --- The Nilness and Semiprimitivity of Commutative Semigroup Algebras --- p.45 / Chapter 3.3 --- Radicals of Cancellative Semigroup Algebras --- p.48 / Chapter 3.3.1 --- Group of Fractions of Cancellative Semigroups --- p.48 / Chapter 3.3.2 --- Jacobson Radical of Cancellative Semigroup Algebras --- p.54 / Chapter 3.3.3 --- Subsemigroups of Polycyclic-by-Finite Groups --- p.57 / Chapter 3.3.4 --- Nilpotent Semigroups --- p.59 / Chapter 3.4 --- Radicals of Algebras of Matrix type --- p.62 / Chapter 3.4.1 --- Properties of Rees Algebras --- p.62 / Chapter 3.4.2 --- Algebras Graded by Elementary Rees Matrix Semigroups --- p.65 / Chapter 3.5 --- Radicals of Inverse Semigroup Algebras --- p.68 / Chapter 3.5.1 --- Properties of Inverse Semigroup Algebras --- p.69 / Chapter 3.5.2 --- Radical of Algebras of Clifford Semigroups --- p.72 / Chapter 3.5.3 --- Semiprimitivity Problems of Inverse Semigroup Algebras --- p.73 / Chapter 3.6 --- Other Semigroup Algebras --- p.76 / Chapter 3.6.1 --- Completely Regular Semigroup Algebras --- p.76 / Chapter 3.6.2 --- Separative Semigroup Algebras --- p.77 / Chapter 3.7 --- Radicals of Pi-semigroup Algebras --- p.80 / Chapter 3.7.1 --- PI-Algebras --- p.80 / Chapter 3.7.2 --- Permutational Property and Algebras of Permutative Semigroups --- p.80 / Chapter 3.7.3 --- Radicals of PI-algebras --- p.82 / Chapter 4 --- Finiteness Conditions on Semigroup Algebras --- p.85 / Chapter 4.1 --- Introduction --- p.85 / Chapter 4.1.1 --- Preliminaries --- p.85 / Chapter 4.1.2 --- Semilattice Graded Rings --- p.86 / Chapter 4.1.3 --- Group Graded Rings --- p.88 / Chapter 4.1.4 --- Groupoid Graded Rings --- p.89 / Chapter 4.1.5 --- Semigroup Graded PI-Algebras --- p.91 / Chapter 4.1.6 --- Application to Semigroup Algebras --- p.92 / Chapter 4.2 --- Semiprime and Goldie Rings --- p.92 / Chapter 4.3 --- Noetherian Semigroup Algebras --- p.99 / Chapter 4.4 --- Descending Chain Conditions --- p.107 / Chapter 4.4.1 --- Artinian Semigroup Graded Rings --- p.107 / Chapter 4.4.2 --- Semilocal Semigroup Algebras --- p.109 / Chapter 5 --- Dimensions and Second Layer Condition on Semigroup Algebras --- p.119 / Chapter 5.1 --- Dimensions --- p.119 / Chapter 5.1.1 --- Gelfand-Kirillov Dimension --- p.119 / Chapter 5.1.2 --- Classical Krull and Krull Dimensions --- p.121 / Chapter 5.2 --- The Growth and the Rank of Semigroups --- p.123 / Chapter 5.3 --- Dimensions on Semigroup Algebras --- p.124 / Chapter 5.4 --- Second Layer Condition --- p.128 / Notations and Abbreviations --- p.132 / Bibliography --- p.133

This research aims to refresh and reinterpret the radical theory of associative rings using the base radical and base semisimple class constructions. It also endeavours to generalise some results about ideals of rings in terms of accessible subrings. A characterisation of accessible subrings is included. By applying the base radical and base semisimple class constructions to many of the known results in established radical theory a number of gaps are uncovered and closed, with the goal of making the theory more accessible to advanced undergraduate and graduate students and mathematicians in related fields, and to open up new areas of investigation. After a literature review and brief reminder of algebra rudiments, the useful properties of accessible subrings and the U and S operators independent from radical class connections are described. The section on accessible subrings illustrates that replacing ideals with accessible subrings is indeed possible for a number of results and demonstrates its usefulness. The traditional radical and semisimple class definitions are included and it is shown that the base radical and base semisimple class constructions are equivalent. Diagrams illustrating the constructions support the definitions. From then on, all radical and semisimple classes mentioned are understood to have the base radical and base semisimple class form. Subject to the constraints of this work, many known results of traditional radical theory are reinterpreted with new proofs, illustrating the potential to simplify the understanding of radical theory using the base radical and base semisimple class constructions. Along with reinterpreting known results, new results emerge giving further insight to radical theory and its intricacies. Accessible subrings and the U and S operators are integrated into the development. The duality between the base radical and base semisimple class constructions is demonstrated in earnest. With a measure of the theory presented, the new constructions are applied to examples and concrete radicals. Context is supported by establishing the relationship between some well-known rings and the radical and related classes of interest. The title of the thesis, Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions, reflects the understanding that reinterpreting the entirety of radical theory is beyond the scope of this work. The conclusion includes an outlook listing further research that time did not allow.

Abstract available in pdf file.

In the budding yeast Saccharomyces cerevisiae, the evolutionarily conserved Target of Rapamycin (TOR) signaling network regulates cell growth in accordance with nutrient and stress conditions. In this work, I present evidence that the TOR complex 1 (TORC1)-interacting proteins Nnk1, Fmp48, Mks1, and Sch9 link TOR to various facets of nitrogen metabolism and mitochondrial function. The Nnk1 kinase controlled nitrogen catabolite repression-sensitive gene expression via Ure2 and Gln3, and physically interacted with the NAD+-linked glutamate dehydrogenase Gdh2 that catalyzes deamination of glutamate to alpha-ketoglutarate and ammonia. In turn, Gdh2 modulated rapamycin sensitivity, was phosphorylated in Nnk1 immune complexes in vitro, and was relocalized to a discrete cytoplasmic focus in response to NNK1 overexpression or respiratory growth. The Fmp48 kinase regulated respiratory function and mitochondrial morphology, while Mks1 linked TORC1 to the mitochondria-to-nucleus retrograde signaling pathway. The Sch9 kinase appeared to act as both an upstream regulator and downstream sensor of mitochondrial function. Loss of Sch9 conferred a respiratory growth defect, a defect in mitochondrial DNA transmission, lower mitochondrial membrane potential, and decreased levels of reactive oxygen species. Conversely, loss of mitochondrial DNA caused loss of Sch9 enrichment at the vacuolar membrane, loss of Sch9 phospho-isoforms, and small cell size suggestive of reduced Sch9 activity. Sch9 also exhibited dynamic relocalization in response to stress, including enrichment at mitochondria under conditions that have previously been shown to induce apoptosis in yeast. Taken together, this work reveals intimate connections between TORC1, nitrogen metabolism, and mitochondrial function, and has implications for the role of TOR in regulating aging, cancer, and other human diseases.

In the budding yeast Saccharomyces cerevisiae, the evolutionarily conserved Target of Rapamycin (TOR) signaling network regulates cell growth in accordance with nutrient and stress conditions. In this work, I present evidence that the TOR complex 1 (TORC1)-interacting proteins Nnk1, Fmp48, Mks1, and Sch9 link TOR to various facets of nitrogen metabolism and mitochondrial function. The Nnk1 kinase controlled nitrogen catabolite repression-sensitive gene expression via Ure2 and Gln3, and physically interacted with the NAD+-linked glutamate dehydrogenase Gdh2 that catalyzes deamination of glutamate to alpha-ketoglutarate and ammonia. In turn, Gdh2 modulated rapamycin sensitivity, was phosphorylated in Nnk1 immune complexes in vitro, and was relocalized to a discrete cytoplasmic focus in response to NNK1 overexpression or respiratory growth. The Fmp48 kinase regulated respiratory function and mitochondrial morphology, while Mks1 linked TORC1 to the mitochondria-to-nucleus retrograde signaling pathway. The Sch9 kinase appeared to act as both an upstream regulator and downstream sensor of mitochondrial function. Loss of Sch9 conferred a respiratory growth defect, a defect in mitochondrial DNA transmission, lower mitochondrial membrane potential, and decreased levels of reactive oxygen species. Conversely, loss of mitochondrial DNA caused loss of Sch9 enrichment at the vacuolar membrane, loss of Sch9 phospho-isoforms, and small cell size suggestive of reduced Sch9 activity. Sch9 also exhibited dynamic relocalization in response to stress, including enrichment at mitochondria under conditions that have previously been shown to induce apoptosis in yeast. Taken together, this work reveals intimate connections between TORC1, nitrogen metabolism, and mitochondrial function, and has implications for the role of TOR in regulating aging, cancer, and other human diseases.