Direct sums of J-rings and radical rings
Let R be a ring, J(R) the Jacobson radical of R and P the set of potent elements of R. We prove that if R satisfies (∗) given x, y in R there exist integers m=m(x,y)>1 and n=n(x,y)>1 such that xmy=xyn and if each x∈R is the sum of a potent element and a nilpotent element, then N and P are idea...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1995-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171295000664 |
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| Summary: | Let R be a ring, J(R) the Jacobson radical of R and P the set of potent
elements of R. We prove that if R satisfies (∗) given x, y in R there exist integers
m=m(x,y)>1 and n=n(x,y)>1 such that xmy=xyn and if each x∈R is
the sum of a potent element and a nilpotent element, then N and P are ideals and R=N⊕P. We also prove that if R satisfies (∗) and if each x∈R has a representation
in the form x=a+u, where a∈P and u∈J(R) ,then P is an ideal and R=J(R)⊕P. |
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| ISSN: | 0161-1712 1687-0425 |