Narrow Results

A $\ast$-ring $R$ is called a nil $\ast$-clean ring if every element of $R$ is a sum of a projection and a nilpotent. Nil $\ast$-clean rings are the $\ast$-version of nil-clean rings introduced by Diesl. This paper is about the nil $\ast$-clean property of rings with emphasis on matrix rings. We show that a $\ast$-ring $R$ is nil $\ast$-clean if and only if $J(R)$ is nil and $R/J(R)$ is nil $\ast$-clean. For a 2-primal $\ast$-ring $R$, with the induced involution given by$(a_{ij}{)}^{\ast }=(a_{ij}^{\ast }{)}^T$, the nil $\ast$-clean property of ${\mathbb{M}}_n(R)$ is completely reduced to that of ${\mathbb{M}}_n({\mathbb{Z}}_2)$. Consequently, ${\mathbb{M}}_n(R)$ is not a nil $\ast$-clean ring for $n=3,4$, and ${\mathbb{M}}_2(R)$ is a nil $\ast$-clean ring if and only if $J(R)$ is nil, $R/J(R)$is a Boolean ring and $a^{\ast }-a\in J(R)$ for all $a\in R$.

In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of $1-pq$, $1-pqp$, $p-pq$ and $p-q$, where $p$, $q$ are projections in different settings, such as ∗-rings, ∗-reducing rings and $C^{\mathrm{\ast }}$-algebras. Moreover, several representations for the core inverses of product, difference and sum of two generalized projections are derived. In particular, a number of examples are given to illustrate our results.