Characterizing Regular Local Rings via Analogues of (*)-properties

Let R be a commutative noetherian local ring. As analogues of (*) -properties introduced by Ghosh, Gupta, and Puthenpurakal, we introduce and study (A) -properties, (B) -properties, and (C) -relations. Using these three notions, we establish a criterion for a local ring to be regular. This recovers and refines the main result of Ghosh, Gupta, and Puthenpurakal and has further applications.