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WIP Abstract Algebra 12

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Nathan Nguyen
2024-10-29 15:58:04 -05:00
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docs/math/abstract-algebra/DF-12-modules-pids.md
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## Section 12.1 The Basic Theory ## Section 12.1 The Basic Theory
**Definition**. The left $R$-module $M$ is said to be a *Noetherian* $R$-module **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.