14 lines
506 B
Markdown
14 lines
506 B
Markdown
# Chapter 12 - Modules over Principal Ideal Domains
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## Section 12.1 The Basic Theory
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**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
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$$
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M_1 \subseteq M_2 \subseteq \ldots
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$$
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there xists some $k \in \mathbb{N}$ such thaht given any $n \in \mathbb{N}$ with $n \geq k$, then $M_n = M_k$.
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**Definition**. A ring $R$ is *Noetherian* if it is Noetherian when viewed as a left $R$-module over itself.
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