交換環理論/貝祖域

定義（貝祖域）:

貝祖域是一個整環，其所有有限生成的理想都是主理想，即由單個元素生成。

命題（每個貝祖域都是一個GCD域）:

設 $R$ 是一個貝祖域。那麼 $R$ 是一個GCD域。

Proof: Given any two elements $a,b\in R$, we may consider the ideal $I=\langle a,b\rangle$ generated by $a$ and $b$. By the definition of Bézout domains, $I=\langle c\rangle$ for at least one $c\in R$ (which is moreover unique up to similarity). Then $\langle c\rangle \leq \langle a\rangle$ and $\langle c\rangle \leq \langle b\rangle$, so that by the characterisation of divisibility by principal ideals, $c$ is a common divisor of $a$ and $b$. Moreover, if $d\in R$ is another common divisor of $a$ and $b$, then $a,b\in \langle d\rangle$, so that $\langle a,b\rangle =\langle c\rangle \leq \langle d\rangle$, so that $d|c$. Hence, $c$ is a greatest common divisor of $a$ and $b$. $\Box$