Question

M and N are Noetherian then their direct sum is also Noetherian

Answer 1

we say that an A-module

A^M

m is noetherian if all of its submodules are finitely generated ,having that definition in mind

assume that M _i are noetherian and let be an ($\bigoplus$ _i L _i)n

Mi increasing sequance of ($\bigoplus$_i L_i)_n submodules in $\bigoplus$ _iM_i L_in then in particular ,is an $\bigoplus$ _iM_i

L_in increasing sequence im M_i and hence stabilises ,M_i that is for

L _in =L_in+1.now set ,N_i some =.... then N=max _i N _iNi

N=max_iN_I

L_in =L_in +1=.....($\bigoplus$_iL_i) stabilises for n>N

($\bigoplus$ _iLi and is equal to $\bigoplus$ iLi where Li=LiNi