This paper is concerned to relationship between the sets of associated primes of the d-local cohomology modules and the ordinary local cohomology modules. Let R be a commutative Noetherian local ring, M an R-module and d, t two integers. We prove that Ass(H t d (M)) = I∈Φ Ass(H t I (M)) whenever H i d (M) = 0 for all i < t and Φ = {I : I is an ideal of R with dim R/I ≤ d}. We give some information about the non-vanishing of the d-local cohomology modules. To be more precise, we prove that H i d (R) = 0 if and only if i = n − d whenever R is a Gorenstein ring of dimension n. This result leads to an example which shows that Ass(H n−d d (R)) is not necessarily a finite set.

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