دکتر خدیجه احمدی آملی

‎Let $R$ be a commutative Noetherian ring with identity‎, ‎$I$ be an ideal of $R$‎, ‎and $M$ be an $R$-module‎. ‎Let $k\geqslant‎ -‎1$ be an arbitrary integer‎. ‎In this paper‎, ‎we introduce the notions of $\Rad_M(I)$ and $\ara_M(I)$‎ ‎as the radical and the arithmetic rank of $I$ with respect to $M$‎, ‎respectively‎. ‎We show that the existence of some sort of regular sequences‎ ‎can be depended on $\dim M/IM$ and so‎, ‎we can get some information about local cohomology modules as well‎. ‎Indeed‎, ‎if $\ara_M(I)=n\geq 1$ and ${(\Supp_{R}(M/IM))}_{>k}=\emptyset$‎ ‎(e.g.‎, ‎if $\dim M/IM=k$)‎, ‎then there exist $n$ elements $x_1‎, ..., ‎x_n$ in $I$‎ ‎which is a poor $k$-regular $M$-sequence and generate an ideal‎ ‎with the same radical as $\Rad_M(I)$ and so‎ ‎$H^i_I(M)\cong H^i_{(x_1‎, ..., ‎x_n)}(M)$ for all $i\in \mathbb{N}_0$‎. ‎As an application‎, ‎we show that $\ara_M(I) \leq \dim M+1$‎, ‎which is a refinement of the inequality $\ara_R(I) \leq \dim R+1$ for modules‎, ‎attributed to Kronecker and Forster‎. ‎Then‎, ‎we prove‎ ‎$\dim M-\dim M/IM \leq \cd(I‎, ‎M) \leq \ara_M(I) \leq \dim M$‎, ‎if $(R‎, ‎\mathfrak{m})$ is a local ring and $IM \neq M$‎.

Let R be a commutative Noetherian ring, M be a finitely generated R-module and a be an ideal of R. For an arbitrary integer k ≥ −1, we introduce the concept of k-projective dimension of M de- noted by k-pdR M . We show that the finite k-projective dimension of M is at least k-depth(a, R) − k -depth(a, M ). As a generalization of the Intersection Theorem, we show that for any finitely generated R-module N , in certain conditions, k-pdR M is nearer upper bound for dimN than pdR M . Finally, if M is k-perfect, dimN ≤ k -gradeM that generalizes the Strong Intersection Theorem.

In this paper, we construct the concept of general  Krasner  hyperring based on the  ring  structures and the left general Krasner hypermodule based on the  module structures.  This study introduces  the  trivial left general Krasner hypermodules and  proves that the  trivial left general Krasner   hypermodules  are  different from left  Krasner   hypermodules. We show that for any given general  Krasner  hyperring $R$ and trivial left general Krasner   hypermodules $A, B, {bf_{R}h}$om$(A, B)$ is a left general Krasner   hypermodule and  ${bf_{R}h}$om$(-, B)$,     $ ({bf_{R}h}$om$(A, -) )$ is an   exact covariant functor (contravariant). Finally, we  show that the category ${bf_{R}GKH}$mod (left  trivial general Krasner hypermodules and all (homomorphisms) is an abelian  category and  trivial left general Krasner hypermodules have  a normal injective resolution.

This paper is concerned with the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules. Let R be a commutative Noetherian ring, I , J be two ideals of R and M be an R-module. Let a 2 ˜W (I, J) and t 2 N0 be such that Extt R(R/a,M) 2 S and Extj R(R/a,Hi I,J (M)) 2 S for all i < t and all j > 0. Then for any submodule N of Ht I,J (M) such that Ext1 R(R/a,N) 2 S, we obtain HomR(R/a,Ht I,J (M)/N) 2 S.