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Let be a commutative noetherian local ring, and let be a finitely generated -module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the conormal module, in this paper, we define the characteristic module and the cocharacteristic module of , and investigate their properties. Our main results include characterizations of Cohen--Macaulay and Gorenstein local rings.