Let \(R\) be a commutative ring with identity. Suppose that \(I\) and \(J\) are ideals of \(R\) such that \(R/I\) and \(R/J\) are Noetherian rings. Prove that \(R/(I \cap J)\) is also a Noetherian ring.
Solution
We regard all quotient rings below as \(R\)-modules through the natural quotient maps.
1. The modules \(R/I\) and \(R/J\) are Noetherian as \(R\)-modules.
An \(R\)-submodule of \(R/I\) is exactly an ideal of the quotient ring \(R/I\). Indeed, the scalar action of \(R\) on \(R/I\) factors through \(R/I\) itself. Since \(R/I\) is Noetherian as a ring, every ascending chain of its ideals stabilizes. Therefore every ascending chain of \(R\)-submodules of \(R/I\) stabilizes, so \(R/I\) is a Noetherian \(R\)-module. The same argument applies to \(R/J\).
2. The direct sum\[ R/I\oplus R/J \]is a Noetherian \(R\)-module.
A finite direct sum of Noetherian modules is Noetherian. For completeness, consider the exact sequence \[ 0\longrightarrow R/I \longrightarrow R/I\oplus R/J \longrightarrow R/J \longrightarrow0, \] where the first map includes the first summand and the second projects onto the second summand. If the submodule and quotient in a short exact sequence are Noetherian, then the middle module is Noetherian: for any ascending chain in the middle module, the intersections with the submodule and the images in the quotient both stabilize, forcing the original chain to stabilize. By 1, both outer modules are Noetherian, so the direct sum is Noetherian.
3. There is an injective \(R\)-linear map\[ R/(I\cap J)\longrightarrow R/I\oplus R/J. \]
Define \[ \phi:R\longrightarrow R/I\oplus R/J, \qquad r\longmapsto(r+I,r+J). \] Its kernel is \[ \ker\phi =I\cap J. \] Therefore the first isomorphism theorem gives an injective \(R\)-linear map \[ \bar\phi:R/(I\cap J)\hookrightarrow R/I\oplus R/J. \]
4. The \(R\)-module \(R/(I\cap J)\) is Noetherian.
By 2, the module \[ R/I\oplus R/J \] is Noetherian. Every submodule of a Noetherian module is Noetherian. By 3, \(R/(I\cap J)\) is isomorphic to an \(R\)-submodule of this direct sum. Hence \(R/(I\cap J)\) is a Noetherian \(R\)-module.
5. The quotient ring \(R/(I\cap J)\) is Noetherian.
The ideals of \[ R/(I\cap J) \] are exactly its \(R\)-submodules, because the \(R\)-action factors through the quotient ring. By 4, every ascending chain of such \(R\)-submodules stabilizes. Therefore every ascending chain of ideals of \(R/(I\cap J)\) stabilizes. Thus \[ R/(I\cap J) \] is a Noetherian ring.