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The integral degree of $A\\subset B$, denoted by ${\\rm d}_A(B)$, is defined as the supremum of the degrees of minimal integral equations of elements of $B$ over $A$. It is an invariant that lies in between ${\\rm d}_K(L)$ and $\\mu_A(B)$, the minimal number of generators of the $A$-module $B$. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if $A\\subset B$ is simple; if $A\\subset B$ is projectiv"},"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":"1507.02120","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-07-08T12:18:42Z","cross_cats_sorted":[],"title_canon_sha256":"121b3f5a72d0d7cadb62bec7a6806b5eb3bf3e866bdf8e5efcf6ee65b3a44483","abstract_canon_sha256":"40035a3b32e2c89bb1fcd463c4660d822ee1c1ece6ffc48e0b227ba29881425a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:16.593399Z","signature_b64":"5vntw10yPl4nUpNKWx2DFssuH8uzb/puF6W9+AuxSZnxm8rxpDfS2m5aIq4ePFR3Z9U4UFBjoSFi8gybL+ocBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cf42bf700811bdffc5732995d7fc4437dfa9602199a3d0c0482006644bf871a","last_reissued_at":"2026-05-18T00:22:16.592939Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:16.592939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the integral degree of integral ring extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Bernat Plans, Francesc Planas-Vilanova, Jos\\'e M. 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