Question

Prove that if M ,N are Noetherian submodule of P then M +N is also Noetherian submodule of P.

Answer 1

 M ⊆ M⊕N is a submodule whose quotient is isomorphic to N. Since M and N are noetherian, so is M⊕N. Since M+N is a quotient of M⊕N, then M +N is Noetherian submodule of P.