A class of principal ideal rings arising from the converse of the Chinese remainder theorem
Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I⊕R/J is a cyclic R-module, then I+J=R. The rings R such that R/I⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings is...
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| Format: | Article |
| Language: | English |
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Wiley
2006-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/19607 |
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| _version_ | 1832566608425385984 |
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| author | David E. Dobbs |
| author_facet | David E. Dobbs |
| author_sort | David E. Dobbs |
| collection | DOAJ |
| description | Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I⊕R/J is a cyclic R-module,
then I+J=R. The rings R such that R/I⊕R/J is a cyclic
R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings:
fields, rings isomorphic to K×L for the fields K and L, and special principal ideal rings (R,M) such that M2=0. |
| format | Article |
| id | doaj-art-44dcdb2cda1241e28e96f31a44e9ec2c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2006-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-44dcdb2cda1241e28e96f31a44e9ec2c2025-02-03T01:03:43ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252006-01-01200610.1155/IJMMS/2006/1960719607A class of principal ideal rings arising from the converse of the Chinese remainder theoremDavid E. Dobbs0Department of Mathematics, University of Tennessee, Knoxville 37996-1300, TN, USALet R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I⊕R/J is a cyclic R-module, then I+J=R. The rings R such that R/I⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K×L for the fields K and L, and special principal ideal rings (R,M) such that M2=0.http://dx.doi.org/10.1155/IJMMS/2006/19607 |
| spellingShingle | David E. Dobbs A class of principal ideal rings arising from the converse of the Chinese remainder theorem International Journal of Mathematics and Mathematical Sciences |
| title | A class of principal ideal rings arising from the converse of the
Chinese remainder theorem |
| title_full | A class of principal ideal rings arising from the converse of the
Chinese remainder theorem |
| title_fullStr | A class of principal ideal rings arising from the converse of the
Chinese remainder theorem |
| title_full_unstemmed | A class of principal ideal rings arising from the converse of the
Chinese remainder theorem |
| title_short | A class of principal ideal rings arising from the converse of the
Chinese remainder theorem |
| title_sort | class of principal ideal rings arising from the converse of the chinese remainder theorem |
| url | http://dx.doi.org/10.1155/IJMMS/2006/19607 |
| work_keys_str_mv | AT davidedobbs aclassofprincipalidealringsarisingfromtheconverseofthechineseremaindertheorem AT davidedobbs classofprincipalidealringsarisingfromtheconverseofthechineseremaindertheorem |