The paper shows an extension of the Lichtenbaum–Hartshorne Vanishing theorem for local cohomology modules with respect to a pair of ideals. We also study the attached primes of top local cohomology module [Formula: see text] where [Formula: see text] In the case, where [Formula: see text] we show that [Formula: see text]

We study basic properties of the generalized ideal transforms DI (M, N) and the set of associated primes of the modules RiDI (M, N).

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Let R be a commutative Noetherian ring and Φ be a system of ideals of R. In this paper, we study the annihilators and the set of attached prime ideals of top generalized local cohomology modules with respect to a system of ideals.