
theorem maxirr2:
for R being PIDomain,
    a being non zero Element of R
holds {a}-Ideal is maximal iff a is irreducible
proof
let R be PIDomain, a be non zero Element of R;
set S = {a}-Ideal;
now assume AS: a is irreducible;
  now let J be Ideal of R;
    assume H0: S c= J;
    J is principal by IDEAL_1:def 28;
    then consider b being Element of R such that
    H1: J = {b}-Ideal;
    now per cases by AS,H0,H1,div1;
    case b is Unit of R;
       then J = [#] R by H1,div2;
       hence J is non proper;
       end;
    case b is_associated_to a;
       hence S = J by H1,div1;
       end;
    end;
    hence J = S or J is non proper;
    end;
  then B: S is quasi-maximal by RING_1:def 3;
  now assume S is non proper;
    then S = the carrier of R by SUBSET_1:def 6;
    then a is unital by div0;
    hence contradiction by AS;
    end;
  hence S is maximal by B;
  end;
hence thesis by maxirr;
end;
