@nprofile1q... - if the rings are "too infinite", like the ring of continuous real-valued functions on the real interval, the sum in the definition of the Hasse-Weil zeta function doesn't converge and we should probably leave it alone.

For finite boolean rings it's not so bad! For the 2-element ring with 'and' as times and 'exor' as plus, the zeta function is

1/(1 - 2ˢ).

(Whoops - just noticed a pile of minus sign typos I need to fix. Thanks!)