The Jacobson Radical turns out to be important in proving the BKS
Theorem regarding noncontextual hidden variables in Quantum Theory,
distinguished from Bell's Theorem on local realism hidden variables.
These involves deep properties of Quantum Entanglement among other
See Metod Saniga (Astronomical Institute Slovak Republic) and Michel
Planat (Institut Femto-St CNRS France) and their numerous papers in
arXiv (respectively 35 and 38 papers), including especially their two
"The projective line over the finite quotient ring GF(2)[x]/<x^3 - x>
and quantum entanglement I: Theoretical background," arXiv: quant-ph/
0603051 v2 6 Jun 2006, and their paper II published in 2007 in
Theoretical and Mathematical Physics 151 (2007) as well as quant-ph/
0603206 in arXiv. The second paper goes much more deeply into the BKS
Theorem and their proof of it.
"BKS" refers to Bell-Kochen-Specker.
This is past my bedtime for now, but I'll try to continue later.