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AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We prove that if AG(M) is a tree, then either AG(M) is a s"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.06367","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-01-24T11:24:34Z","cross_cats_sorted":[],"title_canon_sha256":"5cd5c0644abdb3f13764fbaa8a0a1fcda62d298aae2d620495a32fa8e7ad5168","abstract_canon_sha256":"df2830aa03335a403840728a9af120f2f46f9f2ce609039ec459a9a806beca79"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:04.620907Z","signature_b64":"1D1k3Qp34ouVvaiLa8IyeqwpA3wU8/jkLG1nkIKnzaO5vJDoNh1IZIqCGIgsar60xaHnBY+P/NCK73GGRfJBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"796b6f6ad737711275c8f8d8277bd75c8a61f62b2ace0bba6794708a357d9d8c","last_reissued_at":"2026-05-18T01:22:04.620477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:04.620477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The annihilating-submodule graph of modules over commutative rings II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Habibollah Ansari-Toroghy, Shokoufeh Habibi","submitted_at":"2016-01-24T11:24:34Z","abstract_excerpt":"Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). 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