Let $R$ be a commutative ring with nonzero identity, let $ℐ\left(ℛ\right)$ be the set of all ideals of $R$ and $\delta :ℐ\left(ℛ\right)\to ℐ\left(ℛ\right)$ an expansion of ideals of $R$ defined by $I\mapsto \delta \left(I\right)$. We introduce the concept of $(\delta ,2)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(\delta ,2)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then $a^2\in I$ or $b^2\in \delta \left(I\right)$. Our purpose is to extend the concept of $2$-ideals to $(\delta ,2)$-primary ideals of commutative rings. Then we investigate the basic properties of $(\delta ,2)$-primary ideals and also discuss the relations among $(\delta ,2)$-primary, $\delta$-primary...

When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $𝔪$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., $\mathrm{reg}\left(I^m\right)=dm+e$ for $m\gg 0$ and some integers $d$, $e$. This motivates writing $\mathrm{reg}\left(I^m\right)=dm+e_m$ for every $m\ge 0$. The sequence $e_m$, called the of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after proving...

Recently, motivated by Anderson, Dumitrescu’s $S$-finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of $S$-coherent rings, which is the $S$-version of coherent rings. Let $R={⨁}_{\alpha \in G}{R}_{\alpha }$ be a commutative ring with unity graded by an arbitrary commutative monoid $G$, and $S$ a multiplicatively closed subset of nonzero homogeneous elements of $R$. We define $R$ to be graded-$S$-coherent ring if every finitely generated homogeneous ideal of $R$ is $S$-finitely presented. The purpose of this paper is to give...

We say that an ideal I on ${}_{\kappa }\lambda$ is semiproper if the corresponding poset ${\mathbb{P}}_I$ is semiproper. In this paper we investigate properties of semiproper ideals on ${}_{\kappa }\lambda$.

Let $A$ be a commutative ring and $𝒮$ a multiplicative system of ideals. We say that $A$ is $𝒮$-Noetherian, if for each ideal $Q$ of $A$, there exist $I\in 𝒮$ and a finitely generated ideal $F\subseteq Q$ such that $IQ\subseteq F$. In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

For each squarefree monomial ideal $I\subset S=k[x_1,...,x_n]$, we associate a simple finite graph $G_I$ by using the first linear syzygies of $I$. The nodes of $G_I$ are the generators of $I$, and two vertices $u_i$ and $u_j$ are adjacent if there exist variables $x,y$ such that $x{u}_i=y{u}_j$. In the cases, where $G_I$ is a cycle or a tree, we show that $I$ has a linear resolution if and only if $I$ has linear quotients and if and only if $I$ is variable-decomposable. In addition, with the same assumption on $G_I$, we characterize all squarefree monomial ideals...

Let $(R,𝔪)$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq {(x_1,...,x_n)}^2$ is a graded ideal of a polynomial ring $S=K[x_1,...,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ of the first syzygy module ${\mathrm{Syz}}_1\left(𝔪\right)$ of $𝔪$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $\left(I^{k+1}{:}_{R}I\right)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $\left(I^{(k+1)}{:}_R{I}^{\left(1\right)}\right)=I^{\left(k\right)}$ for all $k$, where $I^{\left(k\right)}$ denotes the $k$th symbolic power of $I$. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

Let $G$ be a finite abelian group with identity element $1_G$ and $L={⨁}_{g\in G}{L}^g$ be an infinite dimensional $G$-homogeneous vector space over a field of characteristic $0$. Let $E=E\left(L\right)$ be the Grassmann algebra generated by $L$. It follows that $E$ is a $G$-graded algebra. Let $\left|G\right|$ be odd, then we prove that in order to describe any ideal of $G$-graded identities of $E$ it is sufficient to deal with $G^{\text{'}}$-grading, where $|G^{\text{'}}|\le |G|$, ${dim}_F{L}^{1_{G^{\text{'}}}}=\infty$ and ${dim}_F{L}^{g^{\text{'}}}<\infty$ if $g^{\text{'}}\ne 1_{G^{\text{'}}}$. In the same spirit of the case $\left|G\right|$ odd, if $\left|G\right|$ is even it is sufficient to study only those $G$-gradings...

A nonzero $R$-module $M$ is atomic if for each two nonzero elements $a,b$ in $M$, both cyclic submodules $Ra$ and $Rb$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$-space $X$, the factor rings $C(X,\mathbb{Z})/C_F(X,\mathbb{Z})$ and $C_c\left(X\right)/C_F\left(X\right)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$, the factor ring ${\mathbb{Z}}^X/{\mathbb{Z}}^{\left(X\right)}$ has no atomic ideal. Another result is that for each infinite...

Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only...

Let $A\left(X\right)$ denote a subalgebra of $C\left(X\right)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A\left(X\right)$, we generalize the notion of $z_A^{\beta }$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C\left(X\right)$...

Let $f_i$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_i$ such that $\sum g_i{f}_i=1$. The effective versions of this result bound the degrees of the $g_i$ in terms of the degrees of the $f_j$. The aim of this paper is to generalize this to the case when the $f_i$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.