Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$-ideal of $R$ if for $a,b\in R$, $ab\in I$ and $a\notin \sqrt{0}$ imply $b\in I$. We give a new generalization of the concept of $n$-ideals by defining a proper ideal $I$ of $R$ to be a semi $n$-ideal if whenever $a\in R$ is such that $a^2\in I$, then $a\in \sqrt{0}$ or $a\in I$. We give some examples of semi $n$-ideal and investigate semi $n$-ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new class of...

The notion of a d-sequence in Commutative Algebra was introduced by Craig Huneke, while the notion of a sequence of linear type was introduced by Douglas Costa. Both types of sequences generate ideals of linear type. In this paper we study another type of sequences, that we call c-sequences. They also generate ideals of linear type. We show that c-sequences are in between d-sequences and sequences of linear type and that the initial subsequences of c-sequences are c-sequences. Finally we prove a...

Soient $k$ un corps commutatif et $I=(p_1,\cdots ,p_m)k_n\left[X\right]$ un idéal de l’anneau des polynômes $k[X_1,\cdots ,X_n]$ (éventuellement $I=k_n\left[X\right]$). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme $p$, élément de la clôture intégrale $\overline{I}$ de l’idéal $I$, on a une représentation$$p^m=\sum _{1\le i\le m}{p}_i{q}_i,\text{avec}maxdeg\left(q_i{p}_i\right)\le mdegp+m{d}_1\cdots d_m,$$où $d_i=deg{p}_i,1\le i\le m$.

Let $R$ be a commutative ring with identity and $A\left(R\right)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph $\mathrm{SAG}\left(R\right)$ with the vertex set $A{\left(R\right)}^{*}=A\left(R\right)\setminus \left\{0\right\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap \mathrm{Ann}\left(J\right)\ne \left(0\right)$ and $J\cap \mathrm{Ann}\left(I\right)\ne \left(0\right)$. In this paper, the perfectness of $\mathrm{SAG}\left(R\right)$ for some classes of rings $R$ is investigated.

Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, $f:R\to S$ be a ring homomorphism, $M$ be an $R$-module, $N$ be an $S$-module, and let $\varphi :M\to N$ be an $R$-homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R{\bowtie }^{f}J$ was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of $\left(R{\bowtie }^{f}J\right)$-module called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi$, and denoted by $M{\bowtie }^{\varphi }JN$. We study some homological properties of the $\left(R{\bowtie }^{f}J\right)$-module $M{\bowtie }^{\varphi }JN$. Among other results,...

The Dedekind-Mertens lemma relates the contents of two polynomials and the content of their product. Recently, Epstein and Shapiro extended this lemma to the case of power series. We review the problem with a special emphasis on the case of power series, give an answer to a question posed by Epstein-Shapiro and investigate extensions of some related results. This note is of expository character and discusses the history of the problem, some examples and announces some new results.

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.

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...