We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization...

Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathrm{𝒜ℱ}$ modules. The main result is that if the $R$-module ${\mathrm{Ext}}_R^t(R/𝔞,M)$ is finite (finitely generated), $H_{𝔞}^i\left(M\right)$ is $𝔞$-cofinite for all $i<t$ and $H_{𝔞}^t\left(M\right)$ is minimax then $H_{𝔞}^t\left(M\right)$ is $𝔞$-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H_{𝔞}^i\left(N\right)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology module...

Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 1$ for all $i$, then the $R$-module $H_I^i\left(M\right)$ is $I$-cominimax for all $i$. In fact, $H_I^i\left(M\right)$ is $I$-cofinite for all $i\ge 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 2$ for all $i<t$, then ${\mathrm{Ext}}_R^j(R/I,H_I^i\left(M\right))$ and ${\mathrm{Hom}}_R(R/I,H_I^t\left(M\right))$ are weakly Laskerian for all $i<t$ and all $j\ge 0$. As a consequence, the set of associated primes of $H_I^i\left(M\right)$ is finite for all $i\ge 0$, whenever $dimR/I\le 2$ and...

Let $I$ and $J$ be ideals of a Noetherian local ring $(R,𝔪)$ and let $M$ be a nonzero finitely generated $R$-module. We study the relation between the vanishing of $H_{I,J}^{dimM}\left(M\right)$ and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian $R$-module $M/JM$ is equal to its integral closure relative to the Artinian $R$-module $H_{I,J}^{dimM}\left(M\right)$.

In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.

The aim of this paper is to discuss the flat covers of injective modules over a Noetherian ring. Let R be a commutative Noetherian ring and let E be an injective R-module. We prove that the flat cover of E is isomorphic to ${\prod }_{p\in At{t}_R\left(E\right)}{T}_p$. As a consequence, we give an answer to Xu’s question [10, 4.4.9]: for a prime ideal p, when does $T_p$ appear in the flat cover of E(R/m̲)?