The Jacobson radical of a ring $R$ is defined to be the intersection of all maximal left ideals in $R$. It turns out that the Jacobson radical is the intersection of all maximal right ideals in $R$ as well, so the Jacobson radical does not depend on whether one considers left or right ideals. In particular, the Jacobson radical of a ring is a two-sided ideal. In fact, there are several characterizations of the Jacobson radical that do not appear to be symmetric with respect to "leftness" and "rightness" including the following.

1. The intersection of all maximal left ideals.

2. $\bigcap\{\textrm{Ann}(M)|M\,\textrm{is a simple left}\,R-\textrm{module}\}$

3. $\{x\in R|1-rx\,\textrm{has a left inverse for each}\,r\in R\}$

4. $\{x\in R|1-rx\,\textrm{has a two-sided inverse for each}\,r\in R\}$