Abstract
This chapter presents the fundamental results on rings and modules. It discusses a theorem that if A is a Noetherian (an Artinian) ring, then every finitely generated A-module is Noetherian (Artinian). In particular, if A is a finitely generated algebra over a commutative Noetherian (Artinian) ring R, then A is right and left Noetherian, (Artinian). The chapter provides an overview on idempotents and direct sum decompositions of ideals. It does not include a primitive idempotent decomposition of an idempotent, nor is it uniquely determined even if it did exist. It also reviews Krull–Schmidt–Azumaya theorem, which proves modules with composition series. The chapter explains that an integral domain R is said to be a Dedekind domain if it satisfies the three conditions: (1) R is a Noetherian ring, (2) R is integrally closed, and (3) any nonzero prime ideal of R is maximal.
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