Jacobson radical

Let $R$ be a ring. The Jacobson radical $J(R)$ is the intersection of all maximal ideals of $R$ (in the commutative case), and more generally can be defined as the intersection of annihilators of all simple right $R$-modules.

In a unital ring, an element $a$ lies in $J(R)$ iff for every $r\in R$ the element $1-ar$ is a unit . The Jacobson radical is always an ideal , and $R/J(R)$ captures the “semisimple part” of $R$.

Examples:

• For $R=\mathbb Z$, one has $J(R)=\{0\}$.
• For a local ring $(R,\mathfrak m)$, $J(R)=\mathfrak m$.
• For $R=\mathbb Z/p^k\mathbb Z$, $J(R)$ is the ideal generated by the class of $p$.