交换环理论/贝祖域

定义（贝祖域）:

贝祖域是一个整环，其所有有限生成的理想都是主理想，即由单个元素生成。

命题（每个贝祖域都是一个GCD域）:

设 $R$ 是一个贝祖域。那么 $R$ 是一个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 $b$. $\Box$