
theorem maxirr:
for R being non degenerated comRing,
    a being non zero Element of R
holds {a}-Ideal is maximal implies a is irreducible
proof
let R be non degenerated comRing,
    a be non zero Element of R;
set S = {a}-Ideal;
assume AS: S is maximal;
B: now let x be Element of R;
   assume B1: x divides a;
   now per cases by div1,B1,AS,RING_1:def 3;
   case S = {x}-Ideal;
     hence x is_associated_to a by div1;
     end;
   case {x}-Ideal is non proper;
     then {x}-Ideal = the carrier of R by SUBSET_1:def 6;
     then x is unital by div0;
     hence x is Unit of R;
     end;
   end;
   hence x is Unit of R or x is_associated_to a;
   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 AS,SUBSET_1:def 6,XBOOLE_0:def 10;
   end;
hence thesis by B;
end;
