A class of rings which are algebric over the integers
A well-known theorem of N. Jacobson states that any periodic associative ring is commutative. Several authors (most notably Kaplansky and Herstein) generalized the periodic polynomial condition and were still able to conclude that the rings under consideration were commutative. (See [3]) In this pap...
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| Format: | Article |
| Language: | English |
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Wiley
1979-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171279000478 |
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| _version_ | 1832546566990200832 |
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| author | Douglas F. Rall |
| author_facet | Douglas F. Rall |
| author_sort | Douglas F. Rall |
| collection | DOAJ |
| description | A well-known theorem of N. Jacobson states that any periodic associative ring is commutative. Several authors (most notably Kaplansky and Herstein) generalized the periodic polynomial condition and were still able to conclude that the rings under consideration were commutative. (See [3]) In this paper we develop a structure theory for a class of rings which properly contains the periodic rings. In particular, an associative ring R is said to be a quasi-anti-integral (QAI) ring if for every a≠0 in R there exist a positive integer k and integers n1,n2,…,nk (all depending on a), so that 0≠n1a=n2a2+…+nkak. In the main theorems of this paper, we show that any QAl-ring is a subdirect sum of prime QAl-rings, which in turn are shown to be left and right orders in division algebras which are algebraic over their prime fields. |
| format | Article |
| id | doaj-art-e8f242f2b9fb4feeb19decf726ffa5a8 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1979-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e8f242f2b9fb4feeb19decf726ffa5a82025-02-03T06:48:07ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251979-01-012462765010.1155/S0161171279000478A class of rings which are algebric over the integersDouglas F. Rall0Department of Mathematics, Furman University, Greenville 29613, South Carolina, USAA well-known theorem of N. Jacobson states that any periodic associative ring is commutative. Several authors (most notably Kaplansky and Herstein) generalized the periodic polynomial condition and were still able to conclude that the rings under consideration were commutative. (See [3]) In this paper we develop a structure theory for a class of rings which properly contains the periodic rings. In particular, an associative ring R is said to be a quasi-anti-integral (QAI) ring if for every a≠0 in R there exist a positive integer k and integers n1,n2,…,nk (all depending on a), so that 0≠n1a=n2a2+…+nkak. In the main theorems of this paper, we show that any QAl-ring is a subdirect sum of prime QAl-rings, which in turn are shown to be left and right orders in division algebras which are algebraic over their prime fields.http://dx.doi.org/10.1155/S0161171279000478anti-integralquasi-anti-integralperiodicprime. |
| spellingShingle | Douglas F. Rall A class of rings which are algebric over the integers International Journal of Mathematics and Mathematical Sciences anti-integral quasi-anti-integral periodic prime. |
| title | A class of rings which are algebric over the integers |
| title_full | A class of rings which are algebric over the integers |
| title_fullStr | A class of rings which are algebric over the integers |
| title_full_unstemmed | A class of rings which are algebric over the integers |
| title_short | A class of rings which are algebric over the integers |
| title_sort | class of rings which are algebric over the integers |
| topic | anti-integral quasi-anti-integral periodic prime. |
| url | http://dx.doi.org/10.1155/S0161171279000478 |
| work_keys_str_mv | AT douglasfrall aclassofringswhicharealgebricovertheintegers AT douglasfrall classofringswhicharealgebricovertheintegers |