---- > [!definition] Definition. ([[Jacobson radical]]) > The **Jacobson radical** $J(R)$ of a [[ring]] $R$ is the intersection of all [[maximal ideal|maximal]] [[ideal|ideals]] of $R$. ^definition > [!basicexample] > - For a [[local ring]] $(R, \mathfrak{m})$, $J(R)=\mathfrak{m}$ (good!) > $J(\mathbb{Z})=(0)$, since $0$ is the only integer [[divides|divisible]] by all prime numbers > - $J(\mathbb{Z})=\bigcap_{\mathfrak{p} \text{ prime}}^{} \langle \mathfrak{p} \rangle=(0)$, [[prime iff maximal for nonzero ideals in PID|see here]] (bad!) ^basic-example > [!equivalence] > $x \in J(R)$ if and only if $1-xy$ is a [[unit]] in $R$ for all $y \in R$. ^equivalence > [!proof] Proof of Equivalence. > $\implies.$ Let $x \in J(R)$. If there is $y \in R$ such that $1-xy$ is not a [[unit]], then $1 \notin \langle 1-xy \rangle$ and [[existence of maximal ideals in commutative rings|hence]] there is some [[maximal ideal]] $\mathfrak{m} \subset R$ satisfying $\langle 1-xy \rangle \subset \mathfrak{m}$. In particular, we have that $1-xy=m$ for some $m \in \mathfrak{m}$. But since $x \in J(R) \subset \mathfrak{m}$, hence $xy \in J(R) \subset \mathfrak{m}$, we have that $1=m+xy$ is a sum of elements of $\mathfrak{m}$ and hence belongs to $\mathfrak{m}$, contradicting maximality of $\mathfrak{m}$ (maximal ideals must be proper). Therefore, $1-xy$ must be a unit for all $y \in R$. > $\impliedby$. Contrapositively suppose $x \notin J(R)$. Then there is some [[maximal ideal]] $\mathfrak{m} \not \ni x$, and thus $\mathfrak{m}+\langle x \rangle=R$,[^1] i.e., $t+xy=1$ for some $t \in \mathfrak{m}$, $y \in R$. Since $t \in \mathfrak{m}$, it must be a unit, hence $t=1-xy$ must not be a [[unit]]. ^proof ---- #### [^1]: $\mathfrak{m}+\langle x \rangle$ is an ideal: if it did not equal $R$, then it would be a proper subset of $R$ containing $\mathfrak{m}$, contradicting maximality of $\mathfrak{m}$. ---- #### 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 > ```