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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Hindawi Limited
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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doaj-44dcdb2cda1241e28e96f31a44e9ec2c2020-11-25T00:11:06ZengHindawi LimitedInternational 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 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
David E. Dobbs |
| 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 |
| author_facet |
David E. Dobbs |
| author_sort |
David E. Dobbs |
| title |
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_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_sort |
class of principal ideal rings arising from the converse of the
chinese remainder theorem |
| publisher |
Hindawi Limited |
| series |
International Journal of Mathematics and Mathematical Sciences |
| issn |
0161-1712 1687-0425 |
| publishDate |
2006-01-01 |
| 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. |
| url |
http://dx.doi.org/10.1155/IJMMS/2006/19607 |
| work_keys_str_mv |
AT davidedobbs aclassofprincipalidealringsarisingfromtheconverseofthechineseremaindertheorem AT davidedobbs classofprincipalidealringsarisingfromtheconverseofthechineseremaindertheorem |
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1725405153054425088 |