Ding, Nanqing; Chen, Jianlong The homological dimensions of simple modules. (English) Zbl 0803.16007 Bull. Aust. Math. Soc. 48, No. 2, 265-274 (1993). Let \(R\) be a ring. If \(R\) is right semiartinian, the weak (respectively, right) global dimension of \(R\) is shown to be equal to the supremum of the flat (respectively, projective) dimensions of the simple right \(R\)- modules. This result was shown earlier by C. Năstăsescu [C. R. Acad. Sci., Paris, Sér. A 268, 685-688 (1969; Zbl 0182.369)]. A right semiartinian ring is a ring with right Gabriel dimension 1. For a nice generalization of the right global dimension result to rings of arbitrary Gabriel dimension, see the paper of J. J. Simón Pinero [Publ. Mat., Barc. 36, 189-195 (1992; Zbl 0773.16003)]. If \(R\) is a commutative coherent ring, then the weak global dimension is shown to be the supremum of the flat (or injective or FP-injective) dimensions of the simple right \(R\)-modules, where \(M\) is FP-injective if \(\text{Ext}(N,M)=0\) for all finitely presented modules \(N\). This result is an easy consequence of Coro. 2.5.10 of S. Glaz [Commutative coherent rings. (Lect. Notes Math. 1371, 1989; Zbl 0745.13004)] and facts about direct limits. If \(R\) is right semiartinian and right coherent, then the weak global dimension of \(R\) is shown to be the supremum of the FP-injective dimensions of the simple right \(R\)-modules. Reviewer: M.L.Teply (Milwaukee) Cited in 9 Documents MSC: 16E10 Homological dimension in associative algebras 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 13D05 Homological dimension and commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators Keywords:flat dimension; global dimension; simple right \(R\)-modules; right semiartinian ring; right Gabriel dimension; coherent ring; weak global dimension; finitely presented modules; FP-injective dimensions Citations:Zbl 0182.369; Zbl 0773.16003; Zbl 0745.13004 PDFBibTeX XMLCite \textit{N. Ding} and \textit{J. Chen}, Bull. Aust. Math. Soc. 48, No. 2, 265--274 (1993; Zbl 0803.16007) Full Text: DOI References: [1] DOI: 10.2307/2035194 · Zbl 0145.04304 · doi:10.2307/2035194 [2] Stenström, Rings of Quotients (1975) · doi:10.1007/978-3-642-66066-5 [3] Fieldhouse, Glasgow Math. J. 13 pp 144– (1972) [4] DOI: 10.1090/S0002-9904-1970-12370-9 · Zbl 0213.04501 · doi:10.1090/S0002-9904-1970-12370-9 [5] Chen, Math. Japon. 36 pp 1123– (1991) [6] DOI: 10.2307/1993382 · Zbl 0100.26602 · doi:10.2307/1993382 [7] Auslander, Nagoya Math. J. 9 pp 67– (1955) · Zbl 0067.27103 · doi:10.1017/S0027763000023291 [8] Anderson, Rings and categories of modules (1973) [9] DOI: 10.2307/1995475 · Zbl 0215.09101 · doi:10.2307/1995475 [10] Kasch, Modules and rings (1982) [11] Teply, Bull. Austral. Math. Soc. 39 pp 215– (1989) [12] DOI: 10.1112/jlms/s2-2.2.323 · Zbl 0194.06602 · doi:10.1112/jlms/s2-2.2.323 [13] DOI: 10.1080/00927878108822627 · Zbl 0458.16015 · doi:10.1080/00927878108822627 [14] DOI: 10.1007/BF01178976 · Zbl 0237.16020 · doi:10.1007/BF01178976 [15] Rotman, An introduction to homological algebra (1979) · Zbl 0441.18018 [16] DOI: 10.1080/00927878708823527 · Zbl 0628.16010 · doi:10.1080/00927878708823527 [17] DOI: 10.2307/1994637 · Zbl 0145.27602 · doi:10.2307/1994637 [18] Fields, Pacific J. Math. 32 pp 345– (1970) · Zbl 0199.35702 · doi:10.2140/pjm.1970.32.345 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.