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Let be a commutative Noetherian local ring, be an ideal of and a finitely generated -module such that and , where is the cohomological dimension of with respect to and is the -grade of . Let be the Matlis dual functor, where is the injective hull of the residue field . We show that there exists the following long exact sequence
where is a non-negative integer, is a regular sequence in on and, for an -module , is the th local cohomology module of with respect...
Let be a commutative ring. The annihilator graph of , denoted by , is the undirected graph with all nonzero zero-divisors of as vertex set, and two distinct vertices and are adjacent if and only if , where for , . In this paper, we characterize all finite commutative rings with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings whose annihilator graphs have clique number , or . Also, we investigate some properties of the annihilator...
Let be an ideal in a commutative Noetherian ring . Then the ideal has the strong persistence property if and only if for all , and has the symbolic strong persistence property if and only if for all , where denotes the th symbolic power of . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial ideal has the...
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