As a common generalization of Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings and generalized Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings,
we say that a ring \documentclass[12pt]{minimal}
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\begin{document}$ R $\end{document} with an involution \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document} is
a generalized Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-ring if for all \documentclass[12pt]{minimal}
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\begin{document}$ x\in R $\end{document} the right annihilator of \documentclass[12pt]{minimal}
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\begin{document}$ x^{n} $\end{document} is
generated by a projection for some positive integer \documentclass[12pt]{minimal}
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\begin{document}$ n $\end{document} depending on \documentclass[12pt]{minimal}
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\begin{document}$ x $\end{document}.
The abelian generalized Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings are closed under finite direct
product. We address the behavior of the generalized Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document} condition
with respect to various constructions and extensions, present
some families of generalized Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings, study connections to the related classes of rings, and
indicate various examples of generalized Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings. Also, we provide
some large classes of finite and infinite-dimensional Banach \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-algebras
that are generalized Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings but neither Rickart \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings nor generalized Baer \documentclass[12pt]{minimal}
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\begin{document}$ \ast $\end{document}-rings.
