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Nathan Nguyen
2024-10-08 21:17:36 -05:00
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docs/math/abstract-algebra/16-rings.md
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# Chapter 16 - Rings
## Section 16.1 - Rings
**Definition**. A nonempty set $S$ is a *ring* if, with two binary operations called addition and multipllication, the following are satisfied:
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Thus, $ab \in P$. By symnetry, assume $a \notin P$. Thus, $b \in P$ by the devinition of a prime ideal, so $b + P = 0 + P$, meaning $R/P$ is an integral domain.
**Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal.
**Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal.