Carlo Toffalori

Decidability of the theory of modules over commutative valuation domains

Abstract We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all V -modules is decidable.

Notes on local o‐minimality

Keywords. Local o-minimality, Strong local o-minimality, !-categoricity. Abstract. We introduce and study some local versions of o-minimality, requir- ing that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighborhood of every point. Motivating ex- amples are the expansions of the order of reals by sine, cosine and other periodic functions.

Superdecomposable pure-injective modules and integral group rings

We prove that if G is a non-trivial finite group, then the integral group ring

Decidability for ℤ[G]‐Modules when G is Cyclic of Prime Order

\mathbb{Z} G

Some Decidability Results for ℤ[G]‐Modules when G is Cyclic of Squarefree Order

possesses a superdecomposable pure-injective module.

Strongly Minimal Modules Over Group Rings

We consider the decision problem for modules over a group ring ℤ[G], where G is a cyclic group of prime order. We show that it reduces to the same problem for a class of certain abelian structures, and we obtain some partial decidability results for this class. Mathematics Subject Classification: 03C60, 03B25.

An undecidability theorem for lattices over group rings

We extend the analysis of the decision problem for modules over a group ring ℤ[G] to the case when G is a cyclic group of squarefree order. We show that separated ℤ[G]-modules have a decidable theory, and we discuss the model theoretic role of these modules within the class of all ℤ[G]-modules. The paper includes a short analysis of the decision problem for the theories of (finitely generated) modules over ℤ[ζm], where m is a positive integer and ζm is a primitive mth root of 1. Mathematics Subject Classification: 03C60, 03B25.

On the undecidability of some classes of abelian-by-finite groups

ABSTRACT We consider modules over a group ring RG where R is a countable Dedekind domain and G is a finite group. We describe the internal structure of those RG-modules which are strongly minimal or satisfy other related model theoretic and algebraic minimality conditions.

Towards the decidability of the theory of modules over finite commutative rings

Abstract Let G be a finite group, T(Z[G]) denote the theory of Z[G]-lattices (i.e. finitely generated Z-torsionfree Z[G]-modules). It is shown that T(Z[G]) is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p4, or p is odd and S is non-cyclic of order p2, or p = 2 and S is a non-cyclic abelian group of order 8 (notice that, in these cases, Z[S]-lattices are a class of wild representation type). More precisely, first we prove that T(Z[S]) is undecidable because it interprets the word problem for finite groups; then we lift undecidability from T(Z[S]) to T(Z[G]).

The decision problem for \({\vec Z}C(p^3)\)-lattices with \(p\) prime

Abstract Let G be a finite group. For every formula o ( υ ) in the language of groups, let K ( G, o ) denote the class of groups H such that o ( H ) is a normal abelian subgroup of H and the quotient group H ;o (H) is isomorphic to G . We show that if G is nilpotent and its order is not square-free, then there exists a formula o(υ) such that the theory of K(G, o) is undecidable.