Communications in Algebra 25(6) (1997)
pp. 1933-1944
A Dichotomy for Finite Length Modules Induced by Local Duality
Markus Schmidmeier
Abstract.
There are several constructions which enable us to pass information from right modules to left modules, like the vector space dual D for modules over an algebra, the elementary duality E for reflexive pure injective modules, the transpose Tr for finitely presented modules and the local duality L for modules with local or perfect endomorphism ring. Now, L can be used to construct the E-dual module, it generalizes D to modules over larger classes of rings, and plays - together with Tr - an important role in the construction of Auslander-Reiten sequences.However, even for finite length modules M neither of the conditions (a) "LM has finite length", (b) "M occurs as the L-dual of some finite length module" or (c) "M is L-reflexive" need to be satisfied. Our main result is
Theorem.
Let R be a semilocal ring and assume that its radical factor is an artin algebra. Let M be a finite length R-module.
- If M is endofinite, that is if M has finite length over its endomorphism ring, (a), (b) and (c) hold.
- If M is not endofinite, neither (a) nor (b) holds. If M is also finitely presented, (c) doesn't hold.
Rosenberg and Zelinsky have shown that indecomposable modules over an artinian PI-ring are finitely generated. As a byproduct we can show a converse of this statement and also get a criterion for the endomorphism ring of a module to be an artinian PI-ring.Mathematical Reviews: 98j:16004
Zentralblatt: 885.16010
Last modified: October 6, 1999, by Markus Schmidmeier