Wakamatsu, Takayoshi On modules with trivial self-extensions. (English) Zbl 0646.16025 J. Algebra 114, No. 1, 106-114 (1988). Let \({}_ BT_ A\) be a bimodule over connected artin algebras A and B with the properties (1) \(T_ A\) and \({}_ BT\) finitely generated (2) \(B=End(T_ A)\) and \(A=End(_ BT)\), and (3) \(Ext\) \(i_ B(T,T)=0=Ext\) \(i_ A(T,T)\) for all integers \(i\geq 1\). It is well known that such modules appear in the tilting theory and in the study of the generalized Nakayama conjecture. The author proves that the projective (injective) dimensions of \(T_ A\) and \({}_ BT\) coincide, if they are finite; in this case \(T_ A\) and \({}_ BT\) have the same number of nonisomorphic indecomposable direct summands. Reviewer: W.Müller Cited in 1 ReviewCited in 65 Documents MSC: 16Gxx Representation theory of associative rings and algebras 16P20 Artinian rings and modules (associative rings and algebras) 16E10 Homological dimension in associative algebras Keywords:balanced modules; tilting modules; connected artin algebras; generalized Nakayama conjecture; indecomposable direct summands PDFBibTeX XMLCite \textit{T. Wakamatsu}, J. Algebra 114, No. 1, 106--114 (1988; Zbl 0646.16025) Full Text: DOI References: [1] Auslander, M.; Reiten, I., On a generalized version of the Nakayama conjecture, (Proc. Amer. Math. Soc., 52 (1975)), 69-74 · Zbl 0337.16004 [2] Happel, D.; Ringel, C. M., Tilted algebras, Trans. Amer. Math. Soc., 274, 399-443 (1982) · Zbl 0503.16024 [3] Tachikawa, H., Quasi-Frobenius Rings and Generalizations, (Lecture Notes in Math., Vol. 351 (1973), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0177.05901 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.