----- > [!proposition] Proposition. ([[prime iff maximal for nonzero ideals in PID]]) > Let $R$ be a [[PID]], and let $I=\langle a \rangle$ be a nonzero [[ideal]] in $R$. Then $I$ is [[prime ideal|prime]] if and only if it is [[maximal ideal|maximal]]. ^proposition > [!proof]- Proof. ([[prime iff maximal for nonzero ideals in PID]]) > > [[maximal ideal|Maximal ideals]] are [[prime ideal|prime]] in every [[ring]], so we just have to show that in this special case $\text{prime} \implies \text{maximal}$. We will use the 'avoiding-quotient' characterizations of prime/maximal ideals. > Let $I=\langle a \rangle$ be a nonzero [[ideal]] in $R$ that is [[prime ideal|prime]] — i.e., $ab \in I \implies a \in I \text{ or }b \in I$ for all $a,b\in R$. Let $J=\langle b \rangle$ be an [[ideal]] of $R$ with $I = \langle a \rangle \subset \langle b \rangle = J$. Now, since $a \in \langle b \rangle$, $a=bc$ for some $c \in R$. Since of course $a=bc \in I$, this means $b \in I$ or $c \in I$. If $c \in I=\langle a \rangle$, then $c=da$ for some $d \in R$, thus $a=bc=bda.$ Since [[cancellation characterization of zero division|cancellation holds]] in [[integral domain|integral domains]] such as $R$, this means $1_{R}=bd \in J$ and hence $J=R$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```