In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the sense that $Z(R_R)=R$, for example, any commutative nil ring is singular: this article Prop. 2.1 Are these (the corollary and the proposition) not inconsistent with each other?
Thanks for any reply!
Lam works almost exclusively with rings with identity on those books.
The identity obviously cannot be in either singular ideal (since its annihilator $\{0\}$ is never essential.)