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Abstract1 . Using some properties of unconditioned M −sequences in dimension > s, we give a finiteness result for the set

Let (R, m) be a Noetherian local ring and M a finitely generated R-module. Let i 0 be an integer. Consider the following property for the Artinian local cohomology module H iIn this paper, we study the property ( * ) of H i m (M) in order to investigate the universal catenaricity of the ring R/ Ann R M and the unmixedness of the ring R/p for certain p in Supp M. We also characterize the property ( * ) for H i m (M) and obtain the associativity formulae for multiplicity of H i m (M) in case where H i m (M) satisfies the property ( * ).