Jacobson radical

Let $𝑅$ be a ring. The Jacobson radical $𝐽 (𝑅)$ is defined as the intersection of the annihilators of all simple modules ring

𝐽 (𝑅) = ⋂ 𝑀 s i m p l e 𝑅 A n n 𝑀

and is thus a two-sided ideal.