An undecidability theorem for lattices over group rings Original Research Article

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]).