Associated and attached primes of local cohomology modules over almost Dedekind domains

In this paper, we prove the Grothendieck's Vanishing Theorem for finitely generated ideals over almost Dedekind domains and show that there exists a non-finitely generated prime ideal over a non-Noetherian almost Dedekind domain that does not satisfy the Grothendieck's Vanishing Theorem. Among the other results, by considering the attached prime ideals and associated prime ideals of representable modules over almost Dedekind domains, we show that if M is a representable R-module and for every non-zero submodule N of M, the ring R/AnnR(N) has acc on d-annihilators, then Att(G(P)(M)) ? {0, P}. Also, if M is representable and for every non-zero submodule N of E(M)/M, the ring R/Ann(R)(N) has acc on d-annihilators, then Att(H-P(1)(M)) ? {0}, where E(M) is an injective envelope of M.