
theorem thpr:
for R being non degenerated comRing,
    a being non zero Element of R
holds a is prime iff {a}-Ideal is prime
proof
let R be non degenerated comRing, a be non zero Element of R;
set S = {a}-Ideal;
A: now assume A0: a is prime;
   now let x,y be Element of R;
     assume A2: x * y in S;
     now per cases by A2,A0,div0;
     case a divides x;
        hence x in S by div0; end;
     case a divides y;
        hence y in S by div0; end;
       end;
     hence x in S or y in S;
     end;
   then B: S is quasi-prime by RING_1:def 1;
   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 A0;
     end;
   hence S is prime by B;
   end;
now assume A0: S is prime;
   B: now let x,y being Element of R;
      assume a divides x * y;
      then x in S or y in S by div0,A0,RING_1:def 1;
      hence a divides x or a divides y by div0;
      end;
   now assume a is unital;
      then {1.R}-Ideal c= S by div0,IDEAL_1:67;
      then the carrier of R c= S by IDEAL_1:51;
      hence contradiction by A0,SUBSET_1:def 6,XBOOLE_0:def 10;
      end;
   hence a is prime by B;
   end;
hence thesis by A;
end;
