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A dichotomy for finite length modules induced by local duality. (English) Zbl 0885.16010

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.

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
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