Noetherian and Artinian Rings. Primary Decomposition

In this final chapter we introduce Noetherian and Artinian rings through chain conditions on ideals. We prove that every ideal in a Noetherian ring is finitely generated and can be decomposed as a finite intersection of primary ideals. Additionally, we prove that A is Noetherian if and only if A \( [\textit{x}_{1}, . . . ,\textit{x}_{\textit{n}}] \) is Noetherian, and that A is Artinian if and only if, up to isomorphism, it is a finite direct sum of local Artinian rings. This result generalizes what we have seen in Chapters 2 and 3, where A was a quotient of the polynomial ring K \( [\textit{x}_{1}, . . . ,\textit{x}_{\textit{n}}] \) .