Lynch's conjecture and attached primes of the top local cohomology modules

Let a be an ideal of a commutative Noetherian ring R and let M be a finitely generated R-module. The purpose of this paper is to establish some new results on Lynch's conjecture and the set of the attached primes of the top local cohomology module H-a(cd(a,M)) (M) for the case that R has prime characteristic and cd(a, M) = araM(a). To prove our results, we establish and use a new relation between the set AttR(H-a(cd(a,M)) (M)) and the M-height of the annihilator of the top local cohomology module H-a(cd(a,M)) (M). Several corollaries of this result are proved. Among other things, we will provide a complete characterization of the attached primes of the top local cohomology module H-a(cd(a,M)) (M) and Rad(AnnR(H-a(cd(a,M)) (M))). Using this, we show that the ideal AnnR(H-a(cd(a,M)) (M)) has M-height zero.