For a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1...

An extension $D\subseteq R$ of integral domains is strongly$t$-compatible (resp., $t$-compatible) if ${\left(IR\right)}^{-1}={\left(I^{-1}R\right)}_v$ (resp., ${\left(IR\right)}_v={\left(I_{v}R\right)}_v)$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D\subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$.

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.

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.

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.

Let ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ be the ring of Gaussian integers modulo $n$. We construct for ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ a cubic mapping graph $\Gamma \left(n\right)$ whose vertex set is all the elements of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ and for which there is a directed edge from $a\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ to $b\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ if $b=a^3$. This article investigates in detail the structure of $\Gamma \left(n\right)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in ${\Gamma }_1\left(n\right)$ is obtained and we also examine when a vertex lies on a $t$-cycle of ${\Gamma }_2\left(n\right)$, where ${\Gamma }_1\left(n\right)$ is induced by all the units of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ while ${\Gamma }_2\left(n\right)$ is induced by all the...

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)?