  We investigate the possibility of constructing Kochen-Specker uncolorable
sets of idempotent matrices whose entries lie in various rings, including the
rational numbers, the integers, and finite fields. Most notably, we show that
there is no Kochen-Specker coloring of the $n \times n$ idempotent integer
matrices for $n \geq 3$, thereby illustrating that Kochen-Specker contextuality
is an inherent feature of pure matrix algebra. We apply this to generalize
recent no-go results on noncommutative spectrum functors, showing that any
contravariant functor from rings to sets (respectively, topological spaces or
locales) that restricts to the Zariski prime spectrum functor for commutative
rings must assign the empty set (respectively, empty space or locale) to the
matrix ring $M_n(R)$ for any integer $n \geq 3$ and any ring $R$. An appendix
by Alexandru Chirvasitu shows that Kochen-Specker colorings of idempotents in
partial subalgebras of $M_3(F)$ for a perfect field $F$ can be extended to
partial algebra morphisms into the algebraic closure of $F$.
