Example of a (non-simple) right $R$-module that is not annihilated by the Jacobson radical $J(R)$.

Question

According to Wikipedia,

the Jacobson radical of a ring $R$ is the ideal consisting of those elements in $R$ that annihilate all simple right $R$-modules. [emphasis mine]

Is there an easy example of a ring $R$ and a (non-simple) right $R$-module that is not annihilated by $J(R)$, the Jacobson radical of $R$?

Answer 1

Any ring $R$ with nonzero Jacobson radical (e.g., $R=\mathbb{Z}/4\mathbb{Z}$) and the regular right module $R_R$.