Narrow Results

The notions of inner-star and star-inner generalized inverses are introduced on the set of all regular elements in a ∗-ring ${ℛ}$. We thus extend the concept of inner-star and star-inner complex matrices. We study properties of these hybrid generalized inverses on the set ${{ℛ}}^{†}$ of all Moore–Penrose invertible elements in ${ℛ}$ and thus generalize some known results. Partial orders that are induced by inner-star and star-inner inverses are introduced on ${{ℛ}}^{†}$, their properties are examined, and their characterizations are presented.

Let ${ℛ}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(ℂ)$, the set of all $m\times n$ matrices over $ℂ$, to the set ${{ℛ}}^{†}$ of all Moore–Penrose invertible elements in ${ℛ}$. We study partial orders on ${{ℛ}}^{†}$ that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart ∗-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert spaces. Properties of these relations are investigated and some known results are thus generalized.

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 $a\in R$.

We associate a simple undirected graph to a ∗-ring $R$ whose vertex set is $V({\mathrm{\Gamma }}_s^{\ast }(R))=\{0\ne a\in R$$|$$r_R(aR)\ne \{0\}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $aR{b}^{\ast }=0$. This graph is called the strong zero-divisor graph of a ∗-ring $R$ and denoted by ${\mathrm{\Gamma }}_s^{\ast }(R)$. In general ${\mathrm{\Gamma }}_s^{\ast }(R)$ is not connected. It is proved that ${\mathrm{\Gamma }}_s^{\ast }(R)$ is connected and $\mathrm{\text{diam}}({\mathrm{\Gamma }}_s^{\ast }(R))\le 3$, when $R$ is a principally quasi-Baer (p.q.-Baer) ∗-ring. We characterize the $gr({\mathrm{\Gamma }}_s^{\ast }(R))$ using the lattice of central projections of a p.q.-Baer ∗-ring $R$. We proved a sort of Beck’s conjecture for the strong zero-divisor graph ${\mathrm{\Gamma }}_s^{\ast }(R)$ of a p.q.-Baer ∗-ring $R$. It is proved that for a p.q.-Baer ∗-ring $R$, ${\mathrm{\Gamma }}_s^{\ast }(R)$ is uniquely complemented.