Related papers: Blowups of hypersurfaces

In this paper we present an ADE-type classification of hypersurfaces of complete regular local rings based on their Cohen-Macaulay type. In order to preform this classification, we show how we can generalize the classical result regarding…

The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis…

This work concerns surjective maps $\varphi\colon R\to S$ of commutative noetherian local rings with kernel generated by a regular sequence that is part of a minimal generating set for the maximal ideal of $R$. The main result provides…

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper show that such rings have rigid homological properties; for instance, they…

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i:…

Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules whose trace equals the maximal ideal. We investigate the…

We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees…

Let $B$ be a Noetherian normal local ring, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime order. Let $A$ be the ring of $G$-invariants of $B$, assume that $A$ is Noetherian. We study the invariant morphism; in…

Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that…

Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of isonoetherian and…

This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient…

Given a local noetherian ring $R$ whose formal completion is integral, we introduce and study the $p$-radical closure $R^\text{prc}$. Roughly speaking, this is the largest purely inseparable $R$-subalgebra inside the formal completion…

The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local…

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension.…

In this paper, the notion of rings with uniformly S-w-Noetherian spectrum is introduced. Several characterizations of rings with uniformly S-w-Noetherian spectrum are given. Actually, we show that a ring R has uniformly S-w-Noetherian…

We introduce a general notion of solution for a Noetherian differential $k$-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions under which such solutions may exist and be…

Let $(A,\mathfrak{m})$ be a complete Cohen-Macaulay local ring. Assume $A$ is not Gorenstein. We say $A$ is a Teter ring if there exists a complete Gorenstein ring $(B,\mathfrak{n})$ with $\dim B = \dim A$ and a surjective map $B…

In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster's celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that…

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if $I$ is a proper ideal of the ring $R=F[t_1,\ldots,t_n]$ of polynomials over a field $F$, then…

This paper explores the structure of quasi-socle ideals I=Q:m^2 in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problem of when Q is a reduction of I and when the…