On semilocal modules and rings |
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Authors: | Christian Lomp |
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Institution: | Heinrich Heine Universit?t , Düsseldorf, D-40225, Germany E-mail: lomp@math.uni-duesseldqrf.de |
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Abstract: | It is well-known that a ring Ris semiperfect if and only if RR (orRR ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (orRR ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal). |
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