
theorem Th27:
  for R,S being Abelian add-associative right_zeroed
  right_complementable associative distributive well-unital non empty
  doubleLoopStr, P being Function of R,S st P is RingIsomorphism & R is
  Noetherian holds S is Noetherian
proof
  let R,S be Abelian add-associative right_zeroed right_complementable
  associative distributive well-unital non empty doubleLoopStr, P be Function
  of R,S;
  assume that
A1: P is RingIsomorphism and
A2: R is Noetherian;
  now
    P is RingEpimorphism by A1;
    then
A3: P is onto;
    let I be Ideal of S;
    P is RingMonomorphism by A1;
    then
A4: P is one-to-one;
    reconsider PI=P".:I as Ideal of R by A1,Th22,Th25;
    PI is finitely_generated by A2;
    then consider F being non empty finite Subset of R such that
A5: P".:I = F-Ideal;
    P is onto by A3;
    then P" = (P qua Function)" by A4,TOPS_2:def 4;
    then P.:(P".:I) = P.:(P"I) by A4,FUNCT_1:85;
    then P.:(P".:I) = I by A3,FUNCT_1:77;
    then I = (P.:F)-Ideal by A1,A5,Th26;
    hence I is finitely_generated;
  end;
  hence thesis;
end;
