We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ${\mathbb{Z}}_3\oplus {\mathbb{Z}}_3$; ${\mathbb{Z}}_3\oplus B$ where $B$ is a Boolean ring; local ring with nil Jacobson radical; $M_2\left({\mathbb{Z}}_2\right)$ or $M_2\left({\mathbb{Z}}_3\right)$; or the ring of a Morita context with zero pairings where the underlying rings are ${\mathbb{Z}}_2$ or ${\mathbb{Z}}_3$.

We present new characterizations of the rings for which every element is a sum of two tripotents and a nilpotent that commute. These extend the results of Z. L. Ying, M. T. Koşan, Y. Zhou (2016) and Y. Zhou (2018).

A right $R$-module $M$ is called $R$-projective provided that it is projective relative to the right $R$-module $R_R$. This paper deals with the rings whose all nonsingular right modules are $R$-projective. For a right nonsingular ring $R$, we prove that $R_R$ is of finite Goldie rank and all nonsingular right $R$-modules are $R$-projective if and only if $R$ is right finitely $\Sigma$-$CS$ and flat right $R$-modules are $R$-projective. Then, $R$-projectivity of the class of nonsingular injective right modules is also considered. Over right...