{"id":"W2048778290","doi":"10.1006/jabr.1999.8119","title":"The Nilpotence of the τ-Closed Prime Radical in Rings with τ-Krull Dimension","year":2000,"lang":"en","type":"article","venue":"Journal of Algebra","topic":"Rings, Modules, and Algebras","field":"Mathematics","cited_by":8,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Manitoba","funders":"","keywords":"Krull dimension; Mathematics; Global dimension; Associated prime; Pure mathematics; Semiprime ring; Prime (order theory); Prime ideal; Radical of a ring; Dimension (graph theory); Dimension theory (algebra); Torsion (gastropod); Discrete mathematics; Algebra over a field; Noetherian; Combinatorics; Principal ideal ring; Commutative ring","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.002349985,0.0006204915,0.001077365,0.003613617,0.003560135,0.00610307,0.001525275,0.001549187,0.005068415],"category_scores_gemma":[0.005647066,0.0008131426,0.0007338746,0.001405921,0.008196433,0.01240068,0.003574695,0.002668688,0.0005509127],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001289404,"about_ca_system_score_gemma":0.0007825076,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0003021,"about_ca_topic_score_gemma":0.000206848,"domain_scores_codex":[0.9988903,0.0002133977,0.00007925765,0.0002896834,0.0002216657,0.0003056245],"domain_scores_gemma":[0.9949809,0.002354488,0.0007948016,0.0003186782,0.0003844755,0.001166798],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0001438745,0.00004338453,0.0005499562,0.00004155488,0.00001144791,0.0002709098,0.0006956054,0.0002548341,0.001495781,0.9930621,0.0003794714,0.003051229],"study_design_scores_gemma":[0.00004560887,0.00006733859,0.000646239,0.00001487138,0.00001774986,0.0004931142,0.0002343416,0.001027716,0.0009866866,0.9950682,0.001367136,0.00003093538],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9053668,0.001294118,0.03938251,0.002022155,0.0004564814,0.00003717921,0.0001928138,0.0002174535,0.05103055],"genre_scores_gemma":[0.9933277,0.0002715696,0.002537936,0.0001116821,0.0003550352,0.00002055309,0.00004472865,0.00002466189,0.00330628],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.00610307,"threshold_uncertainty_score":0.01695555,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01292428970436871,"score_gpt":0.2391712999768164,"score_spread":0.2262470102724477,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}