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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.doi:10.2298/fil1409899s fatcat:lf2uq3dp2zen5k3gev3yru672e