Decidability for Theories of Modules

We are interested here in the first-order theory of /?-modules: in when it is decidable and in when it differs from the theory of finitely generated /^-modules. It is an open problem to characterise those rings over which the common theory TR of /^-modules is decidable, although an answer has been given for some classes of algebras (see [8,11,12]). Here we show that decidability of 7^ is an algebraic property, in that it is invariant under 'effective Morita equivalence' (so depends only on the category of /^-modules)—at least if R is sufficiently decidable. If a ring is of finite representation type then every module over it is a direct sum of finitely presented modules. It follows easily that the theory TR p of the finitely presented modules coincides with TR (if a sentence is satisfied in some module then it is satisfied in a finitely presented module) and is decidable (again, provided the ring is sufficiently decidable). We suspect that the rings of finite representation type are the only artinian rings over which these two theories coincide: as it is, we have been able to show only that over certain algebras not of finite representation type these theories differ. That TR and T f R p are unequal outside finite representation type is not unreasonable since TK[X] and T £[X] are different and one should expect that, for R a A-algebra not of finite representation type, there will be at least one embedding which induces an interpretation of the category of A^J-modules into the category of /^-modules. There are rings not of finite representation type over which the theory of all modules coincides with the theory of finitely presented modules. For instance, let R be a commutative regular ring with the set of principal maximal ideals dense in speci?. It is not difficult to show (cf. §3) that every sentence satisfied in an /^-module is satisfied in a finitely presented i?-module. Morita invariance of decidability actually holds in contexts more general than that of module categories (for an application see [11]). The paper ends with some results on decidability over von Neumann regular rings. As for the background required: we suspect that not many readers will have background both in model theory and in the representation theory of algebras, but considerations of length preclude our including the relevant material here. We refer the reader to [12] or to survey papers and the like. We wish to thank Angus Macintyre, discussions with whom have contributed significantly to the results in this paper.