Anti-isomorphisms of endomorphism rings of locally free modules. (English) Zbl 0655.16016

Let (F,A) denote the unitary left module A over the ring with identity F, and E(F,A) its endomorphism ring. (F,A) is locally free if each finite subset of A is contained in a free summand of A. Assume that (F,A) and (G,B) are locally free, that F is a domain, and each finitely generated left ideal of G is free. Then E(F,A) and E(G,B) are anti-isomorphic if, and only if, A and B are reflexive and mutually dual. If A and B are free, they must be finitely generated. This generalizes results of R. Baer (A, B are vector spaces), [Linear Algebra and Projective Geometry (1952; Zbl 0049.381)] and L. Gewirtzman (A, B are free over principal left ideal domains) [Math. Ann. 159, 278-284 (1965; Zbl 0127.015)]. An example is given in which E(F,A) and E(G,B) are anti- isomorphic and (F,A) and (G,B) are both locally free (but not free) as well as one in which (F,A) is free but (G,B) is locally free (but not free). All anti-isomorphisms of E(F,A) and E(G,B) are determined when (F,A) and (G,B) are torsion-free modules over complete discrete valuation rings.
Reviewer: K.G.Wolfson


16S50 Endomorphism rings; matrix rings
16W20 Automorphisms and endomorphisms
16Gxx Representation theory of associative rings and algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16Dxx Modules, bimodules and ideals in associative algebras
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