Rings Characterized by Their Modules
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Ohio University / OhioLINK
2011
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ndltd-OhioLink-oai-etd.ohiolink.edu-ohiou13082579662021-08-03T05:47:00Z Rings Characterized by Their Modules Holston, Christopher J. Mathematics p-poor modules projectivity domain no p-middle class A ring <i>R</i> is called a right <i>WV</i>-ring if each simple right <i>R</i>-module is injective relative to proper cyclics. A right <i>WV</i>-ring which is not a right <i>V</i>-ring is called a right strictly <i>WV</i>-ring. A right strictly <i>WV</i>-ring <i>R</i> has only three two-sided ideals. If, in addition, <sub><i>R</i></sub><i>J</i>(<i>R</i>) is finitely generated, then it has only three right ideals. It is shown that, given a cyclic right module <i>C</i> over a right <i>WV</i>-ring <i>R</i>, <i>C</i> is noetherian iff every cyclic module in σ[<i>C</i>] is a direct sum of a module which is projective in σ[<i>C</i>] with a module which is either CS or has finite uniform dimension. A module is called p-poor if it is projective only with respect to the semisimple modules. Every ring has a semisimple p-poor module. A ring <i>R</i> is said to have no right p-middle class if every right <i>R</i>-module is either projective or p-poor. It is shown that a ring <i>R</i> with no right p-middle class is isomorphic to <i>S</i> × <i>K</i>, where <i>S</i> is semisimple artinian and <i>K</i> is an indecomposable ring. This <i>K</i> is either zero or exactly one of the following: (i) a semiprimary right SI-ring, (ii) a semiprimary ring with <i>Soc</i>(<i>K</i><sub><i>K</i></sub>)=Z(K<sub>K</sub>)=<i>J</i>(<i>K</i>), (iii) a prime ring with <i>Soc</i>(<i>K</i><sub><i>K</i></sub>)=0, where either <i>J</i>(<i>K</i>)=0 or <i>K</i> is neither left nor right noetherian, or (iv) a ring with infinitely generated essential right socle and <i>J</i>(<i>K</i>)=0. 2011-10-03 English text Ohio University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1308257966 http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1308257966 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. |
| collection |
NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics p-poor modules projectivity domain no p-middle class |
| spellingShingle |
Mathematics p-poor modules projectivity domain no p-middle class Holston, Christopher J. Rings Characterized by Their Modules |
| author |
Holston, Christopher J. |
| author_facet |
Holston, Christopher J. |
| author_sort |
Holston, Christopher J. |
| title |
Rings Characterized by Their Modules |
| title_short |
Rings Characterized by Their Modules |
| title_full |
Rings Characterized by Their Modules |
| title_fullStr |
Rings Characterized by Their Modules |
| title_full_unstemmed |
Rings Characterized by Their Modules |
| title_sort |
rings characterized by their modules |
| publisher |
Ohio University / OhioLINK |
| publishDate |
2011 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1308257966 |
| work_keys_str_mv |
AT holstonchristopherj ringscharacterizedbytheirmodules |
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1719425221529698304 |