Related papers: An upper bound on the reduction number of an ideal

Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule $N$ of $M$ is said to…

We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy…

Let N be a square-free positive integer and let f be a newform of weight 2 on \Gamma_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has…

The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the…

For a set $\cM=\{-\mu,-\mu+1,\ldots, \lambda\}\setminus\{0\}$ with non-negative integers $\lambda,\mu<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(\lambda,\mu;q)$-\emph{covering…

Let $R$ be a commutative ring with identity and let $I$ be a two-generated ideal of $R$. We denote by $\operatorname{SL}_2(R)$ the group of $2 \times 2$ matrices over $R$ with determinant $1$. We study the action of $\operatorname{SL}_2(R)$…

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k},$$ where…

Let R be an associative ring.In the paper we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n > 2, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is…

Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly…

Let $\mathbb N$ be the set of positive integers, and denote by $\lambda(A)=\inf\{t>0:\sum_{a\in A} a^{-t}<\infty\}$ the convergence exponent of $A\subset\mathbb N$. For $0<q\le 1$, $0\le q\le 1$, respectively, the admissible ideals…

Let $p_1 = 2, p_2 = 3,...$ be the sequence of all primes. Let $\epsilon$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \ge 3$ and $a$. We will establish a lower bound for the number of primes…

Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher…

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 =…

In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is…

Let $R$ be a commutative ring with identity and $S$ a multiplicatively closed subset of $R$. This paper aims to introduce the concept of $S$-$n$-ideals as a generalization of $n$-ideals. An ideal $I$ of $R$ disjoint with $S$ is called an…

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of…

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge…

The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next…

In this paper we prove lower bounds for sizes of refutations of unsatisfiable vector Subset Sum instances $\overrightarrow{a}_1 x_1 + \dots + \overrightarrow{a}_n x_n = \overrightarrow{b}$ in the proof system Res(lin$_{\mathbb{F}_q}$) where…

The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$…