Rings with a finite set of nonnilpotents
Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. Conversely, if R is a...
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| Format: | Article |
| Language: | English |
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Wiley
1979-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171279000120 |
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| _version_ | 1832545550565638144 |
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| author | Mohan S. Putcha Adil Yaqub |
| author_facet | Mohan S. Putcha Adil Yaqub |
| author_sort | Mohan S. Putcha |
| collection | DOAJ |
| description | Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θn-ring for some n. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered. |
| format | Article |
| id | doaj-art-f7f74bfa8a5d4f9ea60a74e7e8332a83 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1979-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-f7f74bfa8a5d4f9ea60a74e7e8332a832025-02-03T07:25:35ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251979-01-012112112610.1155/S0161171279000120Rings with a finite set of nonnilpotentsMohan S. Putcha0Adil Yaqub1Department of Mathematics, N.C. State University, Raleigh 27607, N.C., USADepartment of Mathematics, University of California, Santa Barbara 93106, California, USALet R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θn-ring for some n. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.http://dx.doi.org/10.1155/S0161171279000120nil ringdivision ringprimitive ringsemisimple ringsemigroup. |
| spellingShingle | Mohan S. Putcha Adil Yaqub Rings with a finite set of nonnilpotents International Journal of Mathematics and Mathematical Sciences nil ring division ring primitive ring semisimple ring semigroup. |
| title | Rings with a finite set of nonnilpotents |
| title_full | Rings with a finite set of nonnilpotents |
| title_fullStr | Rings with a finite set of nonnilpotents |
| title_full_unstemmed | Rings with a finite set of nonnilpotents |
| title_short | Rings with a finite set of nonnilpotents |
| title_sort | rings with a finite set of nonnilpotents |
| topic | nil ring division ring primitive ring semisimple ring semigroup. |
| url | http://dx.doi.org/10.1155/S0161171279000120 |
| work_keys_str_mv | AT mohansputcha ringswithafinitesetofnonnilpotents AT adilyaqub ringswithafinitesetofnonnilpotents |