Related papers: First-degree prime ideals of composite extensions

Let $\omega$ be a differential 1-form defining an algebraic foliation of codimension 1 in projective space. In this article we use commutative algebra to study the singular locus of $\omega$ through its ideal of definition. Then, we expose…

Let $R$ be a commutative ring with identity. For a finitely generated $R$-module $M$, the notion of associated prime submodules of $M$ is defined. It is shown that this notion inherits most of essential properties of the usual notion of…

A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the…

Let $R$ be a commutative ring with identity and $S \subseteq R$ be a multiplicative set. An ideal $Q$ of $R$ (disjoint from $S$) is said to be $S$-primary if there exists an $s\in S$ such that for all $x,y\in R$ with $xy\in Q$, we have…

For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the…

Bartholdi and Smoktunowicz constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras…

In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.

We show that for every positive integer $k$, there exist $k$ consecutive primes having the property that if any digit of any one of the primes, including any of the infinitely many leading zero digits, is changed, then that prime becomes…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with…

In this note we show that the initial ideal of the annihilator ideal of a generic form is generated by the largest possible monomials in each degree. We also show that the initial ideal with respect to the degree reverse lexicographical…

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta},…

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to…

Let $R$ be a Noetherian ring and $x_1,\ldots,x_t$ a permutable regular sequence of elements in $R$. Then there exists a finite set of primes $\Lambda$ and natural number $C$ so that for all $n_1,\ldots,n_t$ there exists a primary…

Ideals in Leavitt path algebras have been shown to share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (particularly, prime, prime-power, primary,…

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the two-sided ideals of L form a distributive lattice. It is also shown that L is a multiplication ring, that…

In this article we investigate the distribution of prime ideals of residue degree bigger than one across the ideal classes in the class group of a number field $L$. A criterion for the class group of $L$ being generated by the classes of…

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly…

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…