{"paper":{"title":"When an $\\mathscr{S}$-closed submodule is a direct summand","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dejun Wu, Yongduo Wang","submitted_at":"2010-05-02T09:43:30Z","abstract_excerpt":"It is well known that a direct sum of CLS-modules is not, in general, a CLS-module. It is proved that if $M=M_1\\oplus M_2$, where $M_1$ and $M_2$ are CLS-modules such that $M_1$ and $M_2$ are relatively ojective (or $M_1$ is $M_2$-ejective), then $M$ is a CLS-module and some known results are generalized. Tercan [8] proved that if a module $M=M_{1}\\oplus M_{2}$ where $M_{1}$ and $M_{2}$ are CS-modules such that $M_{1}$ is $M_{2}$-injective, then $M$ is a CS-module if and only if $Z_{2}(M)$ is a CS-module. Here we will show that Tercan's claim is not true."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0132","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}