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The homological dimensions of simple modules. (English) Zbl 0803.16007

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.

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
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