Decidability and undecidability

This chapter is concerned with the question: for which rings R is the theory of R -modules decidable? That is, for which rings is there an effective way of deciding whether or not a given sentence is true in every module? The first section begins with some definitions and discussion, for the benefit of those unfamiliar with decidability questions. Then it is noted that since, for instance, the word problem of a ring is interpretable within the theory of its modules, we should impose some minimum conditions on the ring before the question: “is the theory of modules decidable?” becomes a reasonable one. I suggest such a condition: one should be able to tell effectively whether certain systems of linear equations have solutions in the ring. It is noted that a ring of finite representation type has decidable theory of modules (assuming it satisfies this condition). It is also shown that decidability of the theory of modules is preserved by “effective Morita equivalence”. If the word problem for groups is interpretable within the theory of R -modules, then that theory is undecidable. In §2 we use this fact to establish undecidability of the theory of modules over a variety of rings. It is conjectured that any ring of wild representation type has undecidable theory of modules. In the third section, we turn to decidability. Although the first decidability results were proved “with bare hands”, all present results may be achieved by giving an explicit description of the topology on the space of indecomposable pure-injectives. By a result of Ziegler, that is enough to establish decidability of the theory of R -modules.