??so consider the boolean ring presented by N's worth of generators with the pairwise products all 0 ... ??...

a "2-valued" solution to the system of equations is ... either exactly one variable takes the value 1, or exactly none does ... ???...

??now is this "complete" ???... ??no?? ... ??"gap" between "finite" and "cofinite" ?? ...

???so what about "non-principal ideal" here ???....

??what about "ideal class group" (??problematic because of lack of dedekind domain property?? ??or something ??...) and/or "algebraic k-theory" here ???....

???something about ideal as corresponding to zariski-closed subspace ... ??something about the limit singleton as closed but non-open ... ???...

??principal ideal gives open subvariety here ??? ???by closed complement <1 - the generator> ?? ... ???or something ???....

hmmm... ???what about converse??? (??or something??) .... clopen set ... ????.... hmmm... something about "orthogonal congruences" and "idempotent element" ... ?? ...