Structure of weakly periodic rings with potent extended commutators
A well-known theorem of Jacobson (1964, page 217) asserts that a ring R with the property that, for each x in R, there exists an integer n(x)>1 such that xn(x)=x is necessarily commutative. This theorem is generalized to the case of a weakly periodic ring R with a sufficient number of potent exte...
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| Format: | Article |
| Language: | English |
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Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201005051 |
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| _version_ | 1832558202414170112 |
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| author | Adil Yaqub |
| author_facet | Adil Yaqub |
| author_sort | Adil Yaqub |
| collection | DOAJ |
| description | A well-known theorem of Jacobson (1964, page 217) asserts that a
ring R with the property that, for each x in R, there exists
an integer n(x)>1 such that xn(x)=x is necessarily
commutative. This theorem is generalized to the case of a weakly
periodic ring R with a sufficient number of potent extended
commutators. A ring R is called weakly periodic if every x in R can be written in the form x=a+b,
where a is nilpotent and b is potent in the sense
that bn(b)=b for
some integer n(b)>1. It is shown that a weakly periodic ring R in which certain extended commutators are potent must have a nil
commutator ideal and, moreover, the set N of nilpotents forms an ideal which, in fact, coincides with the Jacobson radical of R. |
| format | Article |
| id | doaj-art-ca3ea02c16c2427e9f1ea59817d99bad |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ca3ea02c16c2427e9f1ea59817d99bad2025-02-03T01:33:08ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0125529930410.1155/S0161171201005051Structure of weakly periodic rings with potent extended commutatorsAdil Yaqub0Department of Mathematics, University of California, Santa Barbara 93106, CA, USAA well-known theorem of Jacobson (1964, page 217) asserts that a ring R with the property that, for each x in R, there exists an integer n(x)>1 such that xn(x)=x is necessarily commutative. This theorem is generalized to the case of a weakly periodic ring R with a sufficient number of potent extended commutators. A ring R is called weakly periodic if every x in R can be written in the form x=a+b, where a is nilpotent and b is potent in the sense that bn(b)=b for some integer n(b)>1. It is shown that a weakly periodic ring R in which certain extended commutators are potent must have a nil commutator ideal and, moreover, the set N of nilpotents forms an ideal which, in fact, coincides with the Jacobson radical of R.http://dx.doi.org/10.1155/S0161171201005051 |
| spellingShingle | Adil Yaqub Structure of weakly periodic rings with potent extended commutators International Journal of Mathematics and Mathematical Sciences |
| title | Structure of weakly periodic rings with potent extended commutators |
| title_full | Structure of weakly periodic rings with potent extended commutators |
| title_fullStr | Structure of weakly periodic rings with potent extended commutators |
| title_full_unstemmed | Structure of weakly periodic rings with potent extended commutators |
| title_short | Structure of weakly periodic rings with potent extended commutators |
| title_sort | structure of weakly periodic rings with potent extended commutators |
| url | http://dx.doi.org/10.1155/S0161171201005051 |
| work_keys_str_mv | AT adilyaqub structureofweaklyperiodicringswithpotentextendedcommutators |