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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |