Rigid rings with involution
In this paper, we introduce the property of *-rigid and \alpha -*-rigid for rings with involution (*-rings). These concepts generalize that of rigid and \alpha-rigid properties, respectively. We show that *-rigid is equivalent to *-reduced and \alpha -*-rigid *-rings are semiprime. Also, we show that reduced *-rings are \alpha-*-reversible and \alpha-rigid *-rings are \alpha-*-Armendariz. Moreover, sufficient conditions are given for *-rigid and \alpha-*-rigid *-rings to be rigid and \alpha-rigid, respectively. We give also sufficient condition for \alpha-*-Armendariz *-rings to be \alpha-*-rigid. Furthermore, we investigate the transfer of the properties of *-rigid, \alpha-*-rigid, \alpha-*-reversible and \alpha-*-Armendariz from a *-ring R to some of its extensions; such as polynomial *-ring $R[x]$, localization S^{-1}R of R at S, Laurent polynomial *-ring R[x, x^{-1}], Dorroh extension $D(R,\mathbb{Z})$ and from Ore *-ring to its classical Quotient Q. Finally, we discuss when the skew polynomial *-ring of a *-Baer (*-reversible) *-ring is *-Baer (*-reversible). Also, we show that the polynomial *-ring o f \alpha-*-reversible *-ring R is \alpha-*-reversible if R is *-Armendariz.
ا.د اسامة عبدالله ابورواش, (1-2019)