
theorem alg0:
for R being Ring,
    S being RingExtension of R
for a being Element of S holds Ann_Poly(a,R) = ker(hom_Ext_eval(a,R))
proof
let F be Ring, E being RingExtension of F;
let a be Element of E;
set g = hom_Ext_eval(a,F);
A: now let o be object;
   assume o in Ann_Poly(a,F);
   then consider p being Polynomial of F such that
   A1: o = p & Ext_eval(p,a) = 0.E;
   A2: g.p = 0.E by A1,ALGNUM_1:def 11;
   reconsider b = p as Element of Polynom-Ring F by POLYNOM3:def 10;
   b in {x where x is Element of Polynom-Ring F : g.x = 0.E} by A2;
   hence o in ker(hom_Ext_eval(a,F)) by A1,VECTSP10:def 9;
   end;
now let o be object;
  assume o in ker(hom_Ext_eval(a,F)); then
  o in {x where x is Element of Polynom-Ring F : g.x = 0.E} by VECTSP10:def 9;
  then consider b being Element of Polynom-Ring F such that
  A1: o = b & g.b = 0.E;
  reconsider p = b as Polynomial of F by POLYNOM3:def 10;
  Ext_eval(p,a) = 0.E by A1,ALGNUM_1:def 11;
  hence o in Ann_Poly(a,F) by A1;
  end;
hence thesis by A,TARSKI:2;
end;
