Notes - Rings and Modules HT24, Divisibility
Flashcards
Suppose $R$ is an integral domain and $a, b \in R$. What does it mean that “$a$ divides $b$”, or “$a$ is a factor of $b$”, or “$a \mid b$”?
Suppose $R$ is an integral domain and $a, b, c \in R$. Can you define $c = \text{hcf}(a, b)$, once in terms of divisibility and then in terms of ideals?
- Whenever $d \mid a$ and $d \mid b$, then $d \mid c$.
- If $\{ a, b \} \subseteq dR$, then $cR \subseteq dR$.
Suppose $R$ is an integral domain and $a, b, l \in R$. Can you define $l = \text{lcm}(a, b)$, once in terms of divisibility and then in terms of ideals?
- Whenever $a \mid m$ and $b \mid m$, then $l \mid m$.
- $mR \subseteq aR \cap bR \implies mR \subseteq lR$
Suppose $R$ is an integral domain. Quickly justify that if $a, b \in R$ then if a highest common factor $\text{hcf}(a, b)$ exists, it is unique up to multiplication by a unit.
If $g _ 1, g _ 2$ were two HCFs, then we would have
\[g_1R \subseteq g_2R\]and
\[g_2R \subseteq g_1R\]so
\[g_1R = g_2 R\]but then they are associates, so differ by a unit.
Suppose:
- $R$ is a PID
- $a, b \in R$
Quickly prove that:
- $a$ and $b$ have a highest common factor $h$ (unique up to units but you don’t need to prove this)
- $\exists u, v \in R$ such that $h = ua + vb$
- Let $I = \langle a \rangle + \langle b\rangle = \{ua + vb \mid u \in R, v \in R\}$.
- Since $R$ is a PID, $\exists h \in R$ such that $I = \langle h \rangle$.
- As $h \in I$, $h = ua + vb$ for some $u, v \in R$.
- If $c \mid a$ and $c \mid b$, then $c \mid ua + vb$ so $c \mid h$, hence $h$ is a highest common factor.