In this paper, we introduce the new property $(aR)$, which extends property $(R)$ introduced by Aiena and his collaborators. We investigate the property $(aR)$ in connection with Weyl type theorems, and establish sufficient and necessary conditions for which property $(aR)$ holds. We also study the stability of property $(aR)$ under perturbations by finite rank operators, by nilpotent operators, by quasi-nilpotent operators and by algebraic operators commuting with $T$.