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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| Format: | Article |
| Language: | English |
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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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| _version_ | 1832562673109172224 |
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| author | Xiuzhan Guo |
| author_facet | Xiuzhan Guo |
| author_sort | Xiuzhan Guo |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-6d17a04f1bbc4e5383be72ad83a512ff |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-6d17a04f1bbc4e5383be72ad83a512ff2025-02-03T01:21:59ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251995-01-0118353153410.1155/S0161171295000664Direct sums of J-rings and radical ringsXiuzhan Guo0Department of Mathematics, Claina University of Mining and Technology, Jiangsu, Xuzhou 221008, ChinaLet 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.http://dx.doi.org/10.1155/S0161171295000664periodicpotentor J-ringradical ringdirect sum. |
| spellingShingle | Xiuzhan Guo Direct sums of J-rings and radical rings International Journal of Mathematics and Mathematical Sciences periodic potent or J-ring radical ring direct sum. |
| title | Direct sums of J-rings and radical rings |
| title_full | Direct sums of J-rings and radical rings |
| title_fullStr | Direct sums of J-rings and radical rings |
| title_full_unstemmed | Direct sums of J-rings and radical rings |
| title_short | Direct sums of J-rings and radical rings |
| title_sort | direct sums of j rings and radical rings |
| topic | periodic potent or J-ring radical ring direct sum. |
| url | http://dx.doi.org/10.1155/S0161171295000664 |
| work_keys_str_mv | AT xiuzhanguo directsumsofjringsandradicalrings |