Let d be a k-derivation of k[x,y], where k is a field of characteristic zero. Denote by $\tilde{d}$ the unique extension of d to k(x,y). We prove that if ker d ≠ k, then ker $\tilde{d}$ = (ker d)0, where (ker d)0 is the field of fractions of ker d.

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,...,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ for any ideal $I$ of an arbitrary ring $R$, where ${\omega }_R\left(I\right)=min\{n:I\text{is}\text{an}n\text{-absorbing}\text{ideal}\text{of}R\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions...

We give a simplified approach to the Abhyankar-Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of ${\mathbb{Z}}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{\Gamma }=x\in \mathbb{N}|x+b\mathbb{Z}\in \Gamma \cup 1$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If $H_{\Gamma }$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM...

Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briançon-Skoda theorem for the local ring ${𝒪}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.

Dato un insieme $X$ di $s$ punti nello spazio proiettivo, si costruisce un esplicito ideale canonico $\mathcal{I}$ nel suo anello di coordinate $R$. Si descrivono le componenti omogenee di $\mathcal{I}$ e la struttura della mappa di moltiplicazione $R_{\sigma }\otimes {\mathcal{I}}_{\sigma +1}\to {\mathcal{I}}_{2\sigma +1}$, dove $\sigma =max\left\{i\mid H_X\left(i\right)<s\right\}$. Tra le applicazioni ci sono varie caratterizzazioni di insiemi di punti coomologicamente uniformi, disuguaglianze nelle loro funzioni di Hilbert, il calcolo del primo modulo delle sizigie di $\mathcal{I}$ in casi particolari, una generalizzazione della «trasformata di Gale» a trasformate...

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence $\left(G_n\right)$ of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank ${\omega }^n$.