Covers and Envelopes by Submodules or Quotient-Modules
Let R be a ring, X a class of left R-modules, S the class of submodules of X, and Q the class of quotient-modules of X. It is shown that SQ is precovering (preenveloping) if and only if every injective (projective) left R-module has an X-precover (X-preenvelope). Both epic and monic S-(pre) covers (...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2023/9740245 |
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| author | Yuedi Zeng |
| author_facet | Yuedi Zeng |
| author_sort | Yuedi Zeng |
| collection | DOAJ |
| description | Let R be a ring, X a class of left R-modules, S the class of submodules of X, and Q the class of quotient-modules of X. It is shown that SQ is precovering (preenveloping) if and only if every injective (projective) left R-module has an X-precover (X-preenvelope). Both epic and monic S-(pre) covers (Q-(pre) envelopes) are studied. Moreover, some applications are given. In particular, it is proven that the injective envelope of any projective left R-module is projective if and only if the class of quotient-modules of projective and injective left R-modules is monic preenveloping. |
| format | Article |
| id | doaj-art-e72f4e01994a49a7b99a430924c80cdb |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-e72f4e01994a49a7b99a430924c80cdb2025-08-20T03:26:11ZengWileyJournal of Mathematics2314-47852023-01-01202310.1155/2023/9740245Covers and Envelopes by Submodules or Quotient-ModulesYuedi Zeng0Department of Mathematics and FinanceLet R be a ring, X a class of left R-modules, S the class of submodules of X, and Q the class of quotient-modules of X. It is shown that SQ is precovering (preenveloping) if and only if every injective (projective) left R-module has an X-precover (X-preenvelope). Both epic and monic S-(pre) covers (Q-(pre) envelopes) are studied. Moreover, some applications are given. In particular, it is proven that the injective envelope of any projective left R-module is projective if and only if the class of quotient-modules of projective and injective left R-modules is monic preenveloping.http://dx.doi.org/10.1155/2023/9740245 |
| spellingShingle | Yuedi Zeng Covers and Envelopes by Submodules or Quotient-Modules Journal of Mathematics |
| title | Covers and Envelopes by Submodules or Quotient-Modules |
| title_full | Covers and Envelopes by Submodules or Quotient-Modules |
| title_fullStr | Covers and Envelopes by Submodules or Quotient-Modules |
| title_full_unstemmed | Covers and Envelopes by Submodules or Quotient-Modules |
| title_short | Covers and Envelopes by Submodules or Quotient-Modules |
| title_sort | covers and envelopes by submodules or quotient modules |
| url | http://dx.doi.org/10.1155/2023/9740245 |
| work_keys_str_mv | AT yuedizeng coversandenvelopesbysubmodulesorquotientmodules |