Theorem

$J(R)=\mathcal Z(R_R)$ for principally injective rings

If $R$ is right principally injective, then $J(R)=\mathcal Z(R_R)$ (where $J(R)$ is the Jacobson radical and $\mathcal Z(R_R)$ is the right singular ideal.

Reference(s)

  • W. K. Nicholson and M. F. Yousif. Principally injective rings. (1995) @ Theorem 2.1, Corollary 2.1 p 81