Abstract
This paper is a contribution to the algebraic study of contextuality in quantum theory. As an algebraic analogue of Kochen and Specker's no-hidden-variables result, we investigate rational subrings over which the partial ring of d × d symmetric matrices (d ≥ 3) admits no morphism to a commutative ring, which we view as an "algebraic hidden state." For d = 3, the minimal such ring is shown to be Z[1/6], while for d ≥ 6 the minimal subring is Z itself. The proofs rely on the construction of new sets of integer vectors in dimensions 3 and 6 that have no Kochen-Specker coloring.