We provide several characterizations and investigate properties of Prüfer modules. In fact, we study the connections of such modules with their endomorphism rings. We also prove that for any Prüfer module M, the forcing linearity number of M, fln(M), belongs to {0,1}.

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...

Seguendo le idee presentate nei lavori [1] e [2] si studiano le proprietà dei gruppi di $i$-omotopia per moduli ed omomorfismi di moduli.

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 $(R,𝔪)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $𝔭$ of $M$ such that depth $M=dimR/𝔭$. In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then $I^p$ is a Stanley ideal as well, where $I^p$ is the polarization of $I$.

Let $\ast$ be a star-operation on $R$ and $\ast {}_s$ the finite character star-operation induced by $\ast$. The purpose of this paper is to study when $\ast =v$ or $\ast {}_s=t$. In particular, we prove that if every prime ideal of $R$ is $\ast$-invertible, then $\ast =v$, and that if $R$ is a unique $\ast$-factorable domain, then $R$ is a Krull domain.

Let $C\left(L\right)$ be the ring of real-valued continuous functions on a frame $L$. In this paper, strongly fixed ideals and characterization of maximal ideals of $C\left(L\right)$ which is used with strongly fixed are introduced. In the case of weakly spatial frames this characterization is equivalent to the compactness of frames. Besides, the relation of the two concepts, fixed and strongly fixed ideals of $C\left(L\right)$, is studied particularly in the case of weakly spatial frames. The concept of weakly spatiality is actually weaker than...

In this paper we introduce the class of strongly rectifiable and S-homogeneous modules. We study basic properties of these modules, of their pure and refined submodules, of Hill's modules and we also prove an extension of the second Prüfer's theorem.

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)?