Answer to Question #144112 in Abstract Algebra for anjali g

Question

Direct sum of neotherian module's is neotherian

Answer 1

Assume M = N₁ + N₂ is a module, where N₁, N₂ are noetherian modules.

0 $\to$ N₁ $\xrightarrow[]{i}$ N₁ $\oplus$ N₂ $\xrightarrow[]{j}$ N₂ $\xrightarrow[]{}$ 0

where i(n)=(n,0) and j(n,m)=m. As it is an exact sequence, the module in the middle, i.e. N₁ $\oplus$ N₂, is noetherian if the two other modules are noetherian.