Get quick answers to your questions about the article from our AI researcher chatbot
{'id': 'https://openalex.org/W2075022883', 'doi': 'https://doi.org/10.1090/s0002-9947-1948-0026048-0', 'title': 'Some generalization of quasi-Frobenius algebras', 'display_name': 'Some generalization of quasi-Frobenius algebras', 'publication_year': 1948, 'publication_date': '1948-01-01', 'ids': {'openalex': 'https://openalex.org/W2075022883', 'doi': 'https://doi.org/10.1090/s0002-9947-1948-0026048-0', 'mag': '2075022883'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1090/s0002-9947-1948-0026048-0', 'pdf_url': 'https://www.ams.org/tran/1948-064-01/S0002-9947-1948-0026048-0/S0002-9947-1948-0026048-0.pdf', 'source': {'id': 'https://openalex.org/S60030702', 'display_name': 'Transactions of the American Mathematical Society', 'issn_l': '0002-9947', 'issn': ['0002-9947', '1088-6850'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310315719', 'host_organization_name': 'American Mathematical Society', 'host_organization_lineage': ['https://openalex.org/P4310315719'], 'host_organization_lineage_names': ['American Mathematical Society'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'type': 'article', 'type_crossref': 'journal-article', 'indexed_in': ['crossref'], 'open_access': {'is_oa': True, 'oa_status': 'bronze', 'oa_url': 'https://www.ams.org/tran/1948-064-01/S0002-9947-1948-0026048-0/S0002-9947-1948-0026048-0.pdf', 'any_repository_has_fulltext': False}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5060684119', 'display_name': 'Robert Thrall', 'orcid': None}, 'institutions': [], 'countries': [], 'is_corresponding': True, 'raw_author_name': 'R. M. Thrall', 'raw_affiliation_strings': [], 'affiliations': []}], 'institution_assertions': [], 'countries_distinct_count': 0, 'institutions_distinct_count': 0, 'corresponding_author_ids': ['https://openalex.org/A5060684119'], 'corresponding_institution_ids': [], 'apc_list': None, 'apc_paid': None, 'fwci': 0.0, 'has_fulltext': True, 'fulltext_origin': 'pdf', 'cited_by_count': 91, 'citation_normalized_percentile': {'value': 0.895652, 'is_in_top_1_percent': False, 'is_in_top_10_percent': False}, 'cited_by_percentile_year': {'min': 98, 'max': 99}, 'biblio': {'volume': '64', 'issue': '1', 'first_page': '173', 'last_page': '183'}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T11703', 'display_name': 'Rings, Modules, and Algebras', 'score': 0.9913, 'subfield': {'id': 'https://openalex.org/subfields/2602', 'display_name': 'Algebra and Number Theory'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, 'topics': [{'id': 'https://openalex.org/T11703', 'display_name': 'Rings, Modules, and Algebras', 'score': 0.9913, 'subfield': {'id': 'https://openalex.org/subfields/2602', 'display_name': 'Algebra and Number Theory'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T11673', 'display_name': 'Advanced Topics in Algebra', 'score': 0.9871, 'subfield': {'id': 'https://openalex.org/subfields/2602', 'display_name': 'Algebra and Number Theory'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T10287', 'display_name': 'Algebraic structures and combinatorial models', 'score': 0.9855, 'subfield': {'id': 'https://openalex.org/subfields/2608', 'display_name': 'Geometry and Topology'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}], 'keywords': [{'id': 'https://openalex.org/keywords/indecomposable-module', 'display_name': 'Indecomposable module', 'score': 0.8021198}, {'id': 'https://openalex.org/keywords/faithful-representation', 'display_name': 'Faithful representation', 'score': 0.5232079}, {'id': 'https://openalex.org/keywords/representation', 'display_name': 'Representation', 'score': 0.434769}], 'concepts': [{'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.8064712}, {'id': 'https://openalex.org/C157480366', 'wikidata': 'https://www.wikidata.org/wiki/Q2499534', 'display_name': 'Indecomposable module', 'level': 2, 'score': 0.8021198}, {'id': 'https://openalex.org/C2776639384', 'wikidata': 'https://www.wikidata.org/wiki/Q840396', 'display_name': 'Ideal (ethics)', 'level': 2, 'score': 0.65828276}, {'id': 'https://openalex.org/C200288055', 'wikidata': 'https://www.wikidata.org/wiki/Q2621792', 'display_name': 'Element (criminal law)', 'level': 2, 'score': 0.52582276}, {'id': 'https://openalex.org/C114645060', 'wikidata': 'https://www.wikidata.org/wiki/Q5431294', 'display_name': 'Faithful representation', 'level': 2, 'score': 0.5232079}, {'id': 'https://openalex.org/C202444582', 'wikidata': 'https://www.wikidata.org/wiki/Q837863', 'display_name': 'Pure mathematics', 'level': 1, 'score': 0.46506813}, {'id': 'https://openalex.org/C136119220', 'wikidata': 'https://www.wikidata.org/wiki/Q1000660', 'display_name': 'Algebra over a field', 'level': 2, 'score': 0.45993116}, {'id': 'https://openalex.org/C2776359362', 'wikidata': 'https://www.wikidata.org/wiki/Q2145286', 'display_name': 'Representation (politics)', 'level': 3, 'score': 0.434769}, {'id': 'https://openalex.org/C138885662', 'wikidata': 'https://www.wikidata.org/wiki/Q5891', 'display_name': 'Philosophy', 'level': 0, 'score': 0.0}, {'id': 'https://openalex.org/C111472728', 'wikidata': 'https://www.wikidata.org/wiki/Q9471', 'display_name': 'Epistemology', 'level': 1, 'score': 0.0}, {'id': 'https://openalex.org/C94625758', 'wikidata': 'https://www.wikidata.org/wiki/Q7163', 'display_name': 'Politics', 'level': 2, 'score': 0.0}, {'id': 'https://openalex.org/C17744445', 'wikidata': 'https://www.wikidata.org/wiki/Q36442', 'display_name': 'Political science', 'level': 0, 'score': 0.0}, {'id': 'https://openalex.org/C199539241', 'wikidata': 'https://www.wikidata.org/wiki/Q7748', 'display_name': 'Law', 'level': 1, 'score': 0.0}], 'mesh': [], 'locations_count': 1, 'locations': [{'is_oa': True, 'landing_page_url': 'https://doi.org/10.1090/s0002-9947-1948-0026048-0', 'pdf_url': 'https://www.ams.org/tran/1948-064-01/S0002-9947-1948-0026048-0/S0002-9947-1948-0026048-0.pdf', 'source': {'id': 'https://openalex.org/S60030702', 'display_name': 'Transactions of the American Mathematical Society', 'issn_l': '0002-9947', 'issn': ['0002-9947', '1088-6850'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310315719', 'host_organization_name': 'American Mathematical Society', 'host_organization_lineage': ['https://openalex.org/P4310315719'], 'host_organization_lineage_names': ['American Mathematical Society'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1090/s0002-9947-1948-0026048-0', 'pdf_url': 'https://www.ams.org/tran/1948-064-01/S0002-9947-1948-0026048-0/S0002-9947-1948-0026048-0.pdf', 'source': {'id': 'https://openalex.org/S60030702', 'display_name': 'Transactions of the American Mathematical Society', 'issn_l': '0002-9947', 'issn': ['0002-9947', '1088-6850'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310315719', 'host_organization_name': 'American Mathematical Society', 'host_organization_lineage': ['https://openalex.org/P4310315719'], 'host_organization_lineage_names': ['American Mathematical Society'], 'type': 'journal'}, 'license': None, 'license_id': None, 'version': 'publishedVersion', 'is_accepted': True, 'is_published': True}, 'sustainable_development_goals': [], 'grants': [], 'datasets': [], 'versions': [], 'referenced_works_count': 6, 'referenced_works': ['https://openalex.org/W1585490088', 'https://openalex.org/W2011339472', 'https://openalex.org/W2048641883', 'https://openalex.org/W2322898232', 'https://openalex.org/W2335584542', 'https://openalex.org/W4254321275'], 'related_works': ['https://openalex.org/W4241094354', 'https://openalex.org/W4237107081', 'https://openalex.org/W3105245946', 'https://openalex.org/W2952207848', 'https://openalex.org/W2151315566', 'https://openalex.org/W2147622665', 'https://openalex.org/W2090132053', 'https://openalex.org/W2060542026', 'https://openalex.org/W2044887387', 'https://openalex.org/W1630132514'], 'abstract_inverted_index': {'Introduction.': [0], 'Let': [1], '21': [2, 10, 37, 115, 137, 145, 164, 177, 222], 'be': [3, 128, 149, 181, 226], 'an': [4, 56, 175, 246, 290, 311], 'algebra': [5, 131, 152, 176, 229, 247, 259, 326, 347], 'over': [6], 'a': [7, 18, 30, 39, 117, 129, 150, 166, 182, 195, 227, 233, 307, 324, 350, 379], 'field': [8], 'f.': [9], 'is': [11, 27, 53, 116, 125, 138, 146, 178, 205, 217, 223, 249, 260, 263, 298, 327, 348], 'called': [12], 'quasiFrobenius': [13], '(QF)': [14], 'if': [15, 22, 36, 43, 132, 153, 186, 200, 230], 'it': [16, 231], 'has': [17, 38, 165, 232], 'unit': [19, 40], 'element': [20, 41], 'and': [21, 42, 69, 77, 158, 211, 252, 254, 283, 318, 329, 354, 359], 'every': [23, 44, 133, 154, 159, 345], 'primitive': [24, 31, 155, 160], 'right': [25, 50, 161], 'ideal': [26, 397], 'dual': [28], 'to': [29, 55, 107, 127, 148, 180, 225, 244, 270, 292], 'left': [32, 62, 156], 'ideal;': [33], 'or,': [34], 'equivalently,': [35], 'indecomposable': [45, 57], 'direct': [46, 58, 190, 207], 'constituent': [47, 59, 191], 'of': [48, 60, 66, 85, 90, 136, 163, 174, 188, 192, 209, 266, 274, 278, 333, 337, 369, 387, 396], 'the': [49, 61, 78, 86, 108, 201, 206, 241, 264, 267, 272, 275, 286, 295, 302, 334, 342, 364, 383, 388, 394], 'regular': [51, 63], 'representation': [52, 135, 172, 185], 'equivalent': [54, 312], 'representation.': [64, 237], 'Properties': [65], 'QF': [67, 91, 258], 'algebras': [68, 92, 358, 371], 'rings': [70], 'have': [71], 'been': [72], 'treated': [73], 'by': [74], 'Nakayama,': [75], 'Nesbitt,': [76], 'author': [79], '[2,': [80], '3,': [81], '5,': [82], '6](1).': [83], 'Some': [84, 386], 'most': [87], 'important': [88], 'properties': [89, 332], 'do': [93], 'not': [94, 218], 'characterize': [95], 'these': [96], 'algebras,': [97, 316], 'but': [98], 'occur': [99], 'in': [100, 393], 'more': [101, 299, 366], 'extensive': [102], 'classes.': [103], 'This': [104], 'leads': [105], 'us': [106], 'definitions': [109, 282, 390], 'that': [110, 198, 294, 344], 'follow.': [111], 'Throughout': [112], 'this': [113], 'paper': [114, 269], 'f-algebra': [118], 'with': [119, 213, 374], 'unity': [120], 'element.': [121], 'QF-1': [122, 130, 251, 297], 'Algebra.': [123, 144, 221], '{': [124], 'said': [126, 147, 179, 224], 'faithful': [134, 171, 184, 236], 'its': [139], 'own': [140], 'second': [141], 'commutator.': [142], 'QF-2': [143, 151, 315, 325, 338, 346, 375], 'ideal,': [157, 162], 'unique': [167, 234], 'minimal': [168, 183, 235], 'subideal.': [169], 'A': [170], 'Q3': [173, 193], 'deletion': [187], 'any': [189], 'leaves': [194], 'nonfaithful': [196], 'representation,': [197], 'is,': [199], 'corresponding': [202], 'space': [203], 'V': [204], 'sum': [208], 'VI': [210], 'V2': [212], 'V20': [214], 'then': [215], 'Q31': [216], 'faithful.': [219], 'QF-3': [220, 228, 351, 357, 370], 'We': [238], 'shall': [239], 'use': [240], 'notation': [242], 'QF-12': [243], 'describe': [245], 'which': [248, 309, 323, 362], 'both': [250], 'QF-2,': [253], 'so': [255], 'on.': [256], 'Every': [257], 'QF-123.': [261], 'It': [262], 'purpose': [265], 'present': [268], 'initiate': [271], 'study': [273], 'above': [276, 389], 'classes': [277], 'algebras.': [279, 339, 376], '?1': [280], 'contains': [281, 306, 341], 'notations': [284], 'for': [285, 314], 'paper.': [287], '?2': [288], 'gives': [289, 310], 'example': [291], 'show': [293], 'class': [296, 303, 384], 'general': [300, 367], 'than': [301], 'QF.': [304], '?3': [305], 'theorem': [308], 'definition': [313], '??4': [317], '5': [319], 'discuss': [320], 'conditions': [321], 'under': [322], 'QF,': [328], 'give': [330], 'some': [331], 'Cartan': [335], 'invariants': [336], '?6': [340], 'proof': [343], 'also': [349], 'algebra.': [352], '??7': [353], '8': [355], 'treat': [356], 'include': [360], 'examples': [361], 'illustrate': [363], 'essentially': [365], 'character': [368], 'as': [372], 'compared': [373], '?9': [377], 'treats': [378], 'necessary': [380], 'condition': [381], 'on': [382], 'QF-13.': [385], 'are': [391], 'given': [392], 'language': [395], 'theory': [398]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W2075022883', 'counts_by_year': [{'year': 2023, 'cited_by_count': 1}, {'year': 2021, 'cited_by_count': 2}, {'year': 2020, 'cited_by_count': 1}, {'year': 2019, 'cited_by_count': 2}, {'year': 2017, 'cited_by_count': 1}, {'year': 2016, 'cited_by_count': 2}, {'year': 2012, 'cited_by_count': 1}], 'updated_date': '2024-12-13T16:28:37.880678', 'created_date': '2016-06-24'}