Injectivity, Continuity, and CS Conditions on Group Rings
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2006
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ndltd-OhioLink-oai-etd.ohiolink.edu-ohiou11635210642021-08-03T05:43:40Z Injectivity, Continuity, and CS Conditions on Group Rings Alahmadi, Adel Naif M. Mathematics Group Rings Group Algebras CS Modules and Rings Continuous Modules and Rings Almost Self-injective Modules and Rings <p>Almost self-injective, continuous, quasi-continuous (also known as π-injective), and CS modules are generalizations of injective modules. The main aim of this dissertation is to study almost self-injective, continuous, quasi-continuous, and CS group rings. CS group algebras were initiated by Jain et. al. They showed that <i>K[D<sub>∞</sub>]</i> is a CS group algebra if and only if <i>char(K)</i> ≠ 2. Behn extended this result and showed that if <i>K[G]</i> is a prime group algebra with <i>G</i> polycyclic-by-finite, then <i>K[G]</i> is a CS-ring if and only if <i>G</i> is torsion-free or <i>G ≅ D<sub>∞</sub></i> and <i>char(K)</i> ≠ 2. As a consequence, such a group algebra <i>K[G]</i> is hereditary excepting possibly when <i>K[G]</i> is a domain. We show that if <i>K[G]</i> is a semiprime group algebra of polycyclic-by-finite group <i>G</i> and if <i>K[G]</i> has no direct summands that are domains, then <i>K[G]</i> is a CS-ring if and only if <i>K[G]</i> is hereditary if and only if <i>G/Δ<sup>+</sup>(G)≅ D<sub>∞</sub></i> and <i>char(K)</i> ≠ 2. Furthermore, precise structure of a semiprime CS group algebra <i>K[G]</i> of polycyclic-by-finite group <i>G</i>, when <i>K</i> is algebraically closed, is also provided. Among others, it is shown that (i) every almost self-injective group algebra with no nontrivial idempotents is self-injective, (ii) if <i>G</i> is a torsion group and the group algebra <i>K[G]</i> is quasi-continuous then <i>G</i> is a locally finite group, and (iii) for any group <i>G</i>, if <i>K[G]</i> is continuous then <i>G</i> is locally finite. As a consequence, it follows that a CS group algebra <i>K[G]</i> is continuous if and only if <i>K[G]</i> is principally self-injective if and only if <i>G</i> is locally finite.</p> <p>The properties of endomorphism rings of almost self-injective indecomposable modules have been investigated. It is shown that the endomorphism ring of a uniserial almost self-injective right module is left uniserial. For a domain <i>D</i>, it is proved that <i>D</i> is right almost self-injective if and only if <i>D</i> is a two sided valuation domain.</p> 2006-12-20 English text Ohio University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1163521064 http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1163521064 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
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NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics Group Rings Group Algebras CS Modules and Rings Continuous Modules and Rings Almost Self-injective Modules and Rings |
| spellingShingle |
Mathematics Group Rings Group Algebras CS Modules and Rings Continuous Modules and Rings Almost Self-injective Modules and Rings Alahmadi, Adel Naif M. Injectivity, Continuity, and CS Conditions on Group Rings |
| author |
Alahmadi, Adel Naif M. |
| author_facet |
Alahmadi, Adel Naif M. |
| author_sort |
Alahmadi, Adel Naif M. |
| title |
Injectivity, Continuity, and CS Conditions on Group Rings |
| title_short |
Injectivity, Continuity, and CS Conditions on Group Rings |
| title_full |
Injectivity, Continuity, and CS Conditions on Group Rings |
| title_fullStr |
Injectivity, Continuity, and CS Conditions on Group Rings |
| title_full_unstemmed |
Injectivity, Continuity, and CS Conditions on Group Rings |
| title_sort |
injectivity, continuity, and cs conditions on group rings |
| publisher |
Ohio University / OhioLINK |
| publishDate |
2006 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1163521064 |
| work_keys_str_mv |
AT alahmadiadelnaifm injectivitycontinuityandcsconditionsongrouprings |
| _version_ |
1719424234352017408 |