506 B
506 B
Chapter 12 - Modules over Principal Ideal Domains
Section 12.1 The Basic Theory
Definition. The left $R$-module M is said to be a Noetherian $R$-module if there are no infinitely increasing chains of submodules. That is, given
M_1 \subseteq M_2 \subseteq \ldots
there xists some k \in \mathbb{N} such thaht given any n \in \mathbb{N} with n \geq k, then M_n = M_k.
Definition. A ring R is Noetherian if it is Noetherian when viewed as a left $R$-module over itself.