‎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 M be a right module over a ring R,  M a pre-radical on  [M], and N 2  [M]. In this note we show that if N1; N2 2  [M] are two  M-lifting modules such that Ni is Nj-projective (i; j = 1; 2), then N = N1  N2 is  M-lifting. We investigate when homomorphic image of a  M-lifting module is  M-lifting.

The generalized principal ideal theorem is one of the cornerstones of the dimension theory for the Noetherian rings. For an R-module M, certain submodules of M that play a role analo- gous to that of prime ideals in the ring R are identied. Using this denition, we extend the this theorem to modules.

‎The graph $ AG ( R ) $ {of} a commutative ring $R$ with identity has an edge linking two unique vertices when the product of the vertices equals {the} zero ideal and its vertices are the nonzero annihilating ideals of $R$‎.‎The annihilating-ideal graph with {respect to} an ideal $ ( I ) $, ‎which is {denoted} by $ AG_I ( R ) $‎, ‎has distinct vertices $ K $ and $ J $ that are adjacent if and only if $ KJ\subseteq I $‎. ‎Its vertices are $ \{K\mid KJ\subseteq I\ \text{for some ideal}\ J \ \text{and}\ K$‎, ‎$J \nsubseteq I‎, ‎K\ \text{is a ideal of}\ R\} $‎. ‎The study of the two graphs $ AG_I ( R ) $ and $ AG(R/I) $ and {extending certain} prior findings are two main objectives of this research‎. ‎This studys {among other things‎, ‎the} findings {of this study reveal}‎‎that $ AG_I ( R ) $ is bipartite if and only if $ AG_I ( R ) $ is triangle-free‎.

This paper deals with the concepts of Iµ-g-closed sets, Iµ-semi-open sets and Iµ-semi-preopen sets in ideal topological spaces and investigate several properties of these sets. Some characterizations of Iµ-regular spaces and Iµ-normal spaces are discussed.

‎Let $R$ and $S$ be rings‎, ‎$C= {}_SC_R$ a faithfully semidualizing bimodule‎, ‎and $m$ a non-negative integer‎. ‎In this paper‎, ‎we present some homological properties relative to the class of $R$-modules of $C$-weak injective dimension at most $m$ and the class of $S$-modules of $C$-weak flat dimension at most $m$‎. ‎We introduce and study weak cotorsion modules with respect to $m$ and $C$ by using the class of modules of $C$-weak flat dimension at most $m$.

In this paper, we introduce the concept of the cohomological dimension filtration with respect to a system of ideals.In particular, a characterization of cohomological dimension filtration of a module by the associated prime ideals of its factors is established. As a main result, we provide a necessary and sufficient condition for an ascending chain of submodules of an $\mathfrak{R}$ -module $M$ to be the $\mathrm{cd}$-filtration of $M$, with respect to a system of ideals.

This paper is concerned with the relation between local coho-mology modules defined by a pair of ideals and the Serre subcategories of the category of modules. We characterize the membership of local coho-mology modules in a certain Serre subcategory from lower range or upper range.