
theorem degen:
for R being Ring holds R is degenerated iff the carrier of R = {0.R}
proof
let L be Ring;
now assume AS: L is degenerated;
   B: {0.L} c= the carrier of L;
   now let o be object;
     assume o in the carrier of L; then
     reconsider a = o as Element of L;
     a = a * 1.L .= 0.L by AS;
     hence o in {0.L} by TARSKI:def 1;
     end;
   hence the carrier of L = {0.L} by B,TARSKI:2;
   end;
hence thesis;
end;
