Schmidmeier, Markus A dichotomy for finite length modules induced by local duality. (English) Zbl 0885.16010 Commun. Algebra 25, No. 6, 1933-1944 (1997). Let \(M_R\) be a module with \(S=\text{End}(M_R)\) semilocal, i.e., semisimple modulo its Jacobson radical. The local dual is \(LM={_R\text{Hom}_S}(M,I)\) where \(I=E(_S\overline S)\) is the injective envelope of \(\overline S=S/J(S)\).The main result is Theorem 1 (A dichotomy for finite length modules). Let \(R\) be a semilocal ring and \(\overline R=R/J(R)\) is an Artin algebra. Let \(M_R\) be a module of finite length. Then 1. If \(M\) is endofinite, i.e., if \(M\) has finite length as \(\text{End}(M_R)\)-module, then the following three statements hold: (A) \(LM\) has finite length, (B) \(M\) occurs as \(L\)-dual module of some finite length module, and (C) \(M\) is \(L\)-reflexive. 2. If \(M\) is not endofinite, neither (A) nor (B) holds. If \(M\) is also finitely presented, (C) does not hold. Reviewer: Xue Weimin (Fouzhou) Cited in 1 ReviewCited in 3 Documents MSC: 16D90 Module categories in associative algebras 16D50 Injective modules, self-injective associative rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16L30 Noncommutative local and semilocal rings, perfect rings Keywords:local dualities; injective envelopes; finite length modules; semilocal rings PDFBibTeX XMLCite \textit{M. Schmidmeier}, Commun. Algebra 25, No. 6, 1933--1944 (1997; Zbl 0885.16010) Full Text: DOI References: [1] Anderson F.W., Graduates texts in mathematies 13 (1992) [2] Auslander M., Representation theory of algebras, Proc. Conf. Philadelphia pp 1– (1978) · Zbl 0378.16014 [3] Cohn P. M., Lond. Math. soc. Lect. Notes Series 27 (1977) [4] Ckawley-Boevey W., 168, in: Lond. Math. Soc. Lect. (1992) [5] DOI: 10.1016/0021-8693(80)90082-4 · Zbl 0428.16031 · doi:10.1016/0021-8693(80)90082-4 [6] DOI: 10.2307/2154545 · Zbl 0815.16002 · doi:10.2307/2154545 [7] Jacobson N., Linear algebra (1953) [8] DOI: 10.1090/S0002-9904-1948-09049-8 · Zbl 0032.00701 · doi:10.1090/S0002-9904-1948-09049-8 [9] Krause H., Preprintreihe des S F B 23 pp 1– (1995) [10] DOI: 10.1007/BF01110128 · Zbl 0175.31701 · doi:10.1007/BF01110128 [11] Rosenueg A., Math. Z. 70 pp 372– (1959) [12] Kaplansky I., BuU. AMS (1967) [13] Isaacs I. M., Character Theory of Finite Groups (1976) · Zbl 0337.20005 [14] Müller W., Darstellungstheorie von endlichen Gruppen (1980) 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.