Two properties of the power series ring
For a commutative ring with unity, A, it is proved that the power series ring A〚X〛 is a PF-ring if and only if for any two countable subsets S and T of A such that S⫅annA(T), there exists c∈annA(T) such that bc=b for all b∈S. Also it is proved that a power series ring A〚X〛 is a PP-ring if and only i...
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| Format: | Article |
| Language: | English |
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Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171288000031 |
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| _version_ | 1832563188992835584 |
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| author | H. Al-Ezeh |
| author_facet | H. Al-Ezeh |
| author_sort | H. Al-Ezeh |
| collection | DOAJ |
| description | For a commutative ring with unity, A, it is proved that the power series ring A〚X〛 is a PF-ring if and only if for any two countable subsets S and T of A such that S⫅annA(T), there exists c∈annA(T) such that bc=b for all b∈S. Also it is proved that a power series ring A〚X〛 is a PP-ring if and only if A is a PP-ring in which every increasing chain of idempotents in A has a supremum which is an idempotent. |
| format | Article |
| id | doaj-art-ebc2ff01d6764614ba3325f1f82cf133 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1988-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ebc2ff01d6764614ba3325f1f82cf1332025-02-03T01:20:55ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111191310.1155/S0161171288000031Two properties of the power series ringH. Al-Ezeh0Department of Mathematics, University of Jordan, Amman, JordanFor a commutative ring with unity, A, it is proved that the power series ring A〚X〛 is a PF-ring if and only if for any two countable subsets S and T of A such that S⫅annA(T), there exists c∈annA(T) such that bc=b for all b∈S. Also it is proved that a power series ring A〚X〛 is a PP-ring if and only if A is a PP-ring in which every increasing chain of idempotents in A has a supremum which is an idempotent.http://dx.doi.org/10.1155/S0161171288000031power series ringPP-ringPF-ringflatprojectiveannihilator ideal and idempotent element. |
| spellingShingle | H. Al-Ezeh Two properties of the power series ring International Journal of Mathematics and Mathematical Sciences power series ring PP-ring PF-ring flat projective annihilator ideal and idempotent element. |
| title | Two properties of the power series ring |
| title_full | Two properties of the power series ring |
| title_fullStr | Two properties of the power series ring |
| title_full_unstemmed | Two properties of the power series ring |
| title_short | Two properties of the power series ring |
| title_sort | two properties of the power series ring |
| topic | power series ring PP-ring PF-ring flat projective annihilator ideal and idempotent element. |
| url | http://dx.doi.org/10.1155/S0161171288000031 |
| work_keys_str_mv | AT halezeh twopropertiesofthepowerseriesring |