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Subring Criterion for Rings

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Nathan Nguyen
2024-12-02 08:48:24 -06:00
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docs/math/abstract-algebra/16-rings.md
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@@ -30,6 +30,8 @@ Note that some books impose the condition that $1 \neq 0$. If $1 = 0$, we can sh
**Definition**. A commutative division ring is called a *field*. That is, in a field, every element has an inverse. **Definition**. A commutative division ring is called a *field*. That is, in a field, every element has an inverse.
**Definition**. A subset $S$ of ring $R$ is a *subring* if given any $r, s \in S$, then $rs \in S$ and $r - s \in S$.
## Section 16.2 - Integral Domains and Fields ## Section 16.2 - Integral Domains and Fields
**Definition**. If $R$ is a commutative ring and $r \in R$, then $r$ is said to be a *zero divisor* if there is some nonzero $s \in R$ such that $rs = 0$. **Definition**. If $R$ is a commutative ring and $r \in R$, then $r$ is said to be a *zero divisor* if there is some nonzero $s \in R$ such that $rs = 0$.