The equivalence between the categories of closed and firm modules
by
Leandro Marín
Department of Mathematics, University of Murcia, 30100-Espinardo-Murcia (Spain)

Let R be an associative ring. We consider the categories CMod-R (modules M such that M =~ Hom_R(R, M)) and DMod-R (modules M such that M \otimes_R R =~ M). These categories are known to be equivalent when R² = R and some other cases, but they are not equivalent in general. We shall prove that the categories are equivalent with the canonical functors if and only if the ring R is indistinguishable from a ring S such that S \otimes_S S =~ S. Two rings are indistinguishable if they are 'isomorphic up to some elements with trivial multiplication' (the precise definition was introduced in ), for example two rings with identity are indistinguishable if and only if they are isomorphic and two morita equivalent commutative rings are always indistinguishable.

One of the first problems when studying rings without identity, is to choose an appropiate category of modules that could reflect the properties of the ring. Using our construction is possible to get the ring back from the category, therefore the class of rings S such that S \otimes_S S =~ S seems to be the widest class of rings that can be studied, up to isomorphism, with the categories \sf CMod-S, \sf DMod-S or any other equivalent to those.

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García, J.L.; Marín, L.: Morita theory for associative rings. Preprint.

Date received: November 12, 1998