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Central Queensland University

1.
Chin, Melanie Soo.
Towards a Reinterpretation of the Radical Theory of Associative Rings Using *Base* Radical and *Base* *Semisimple* Class Constructions.

Degree: 2004, Central Queensland University

URL: http://library-resources.cqu.edu.au./thesis/adt-QCQU/public/adt-QCQU20050411.102928

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.

Subjects/Keywords: mathematics; radical; radical theory; ring; ring theory; associative rings; base radical; base semisimple; accessible subrings

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APA (6^{th} Edition):

Chin, M. S. (2004). Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions. (Thesis). Central Queensland University. Retrieved from http://library-resources.cqu.edu.au./thesis/adt-QCQU/public/adt-QCQU20050411.102928

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Chin, Melanie Soo. “Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions.” 2004. Thesis, Central Queensland University. Accessed September 24, 2020. http://library-resources.cqu.edu.au./thesis/adt-QCQU/public/adt-QCQU20050411.102928.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Chin, Melanie Soo. “Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions.” 2004. Web. 24 Sep 2020.

Vancouver:

Chin MS. Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions. [Internet] [Thesis]. Central Queensland University; 2004. [cited 2020 Sep 24]. Available from: http://library-resources.cqu.edu.au./thesis/adt-QCQU/public/adt-QCQU20050411.102928.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Chin MS. Towards a Reinterpretation of the Radical Theory of Associative Rings Using Base Radical and Base Semisimple Class Constructions. [Thesis]. Central Queensland University; 2004. Available from: http://library-resources.cqu.edu.au./thesis/adt-QCQU/public/adt-QCQU20050411.102928

Not specified: Masters Thesis or Doctoral Dissertation