Morita equivalence based on contexts for various categories of modules over associative rings Original Research Article

In this paper we consider the subcategories CMod-R (M set membership, variant MOD-R s.t. M not, vert, similar- HomR(R, M)) and DMod-R (M set membership, variant MOD-R s.t. M circle times operatorR R not, vert, similar- R) of the category of all right R-modules, MOD-R, for an associative ring R, possibly without identity. If R and S are associative rings and we have a Morita context between R and S with epimorphic pairings, it can be deduced from [6, 8] that the induced functors provide equivalences CMod-R not, vert, similar- CMod-S R-CMod not, vert, similar- S-CMod, DMod-R not, vert, similar- DMod-S R-DMod not, vert, similar- S-DMod. We find hypotheses weaker than the surjectivity that let us prove also a converse of this result. As a consequence, we give an example of a ring R such that CMod-R is not equivalent to DMod-R.