Catchup 7

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Indigo5684
2025-09-30 13:19:25 -05:00
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**Definition**. Given a commutative ring $R$ with identity, and $r \in R$, the set
$$
<a> = (r)R = \{ ar : r \in R \}
\langle a \rangle = (r)R = \{ ar : r \in R \}
$$
is an ideal in $R$. Specifically, $<a>$ is a *principal ideal*.
is an ideal in $R$. Specifically, $\langle a \rangle$ is a *principal ideal*.
**Example**. Theorem 16.25. Every ideal in $\mathbb{Z}$ is a principal ideal.