Related papers: Test elements in equal characteristic semianalytic…

Let $k$ be a field of characteristic $p>0$ and $R$ be a subalgebra of $k[X]=k[x_1,...,x_n]$. Let $J(R)$ be the ideal in $k[X]$ defined by $J(R)\Omega_{k[X]/k}^n=k[X]\Omega_{R/k}^n$. It is shown that if it is a principal ideal then $J(R)^q$…

It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates $\mathrm{Ext}^{d+1}_{R}(-,-)$, where $d$ is the Krull…

We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the…

Continuing ideas of a recent preprint of Schwede arXiv:0906.4313 we study test ideals by viewing them as minimal objects in a certain class of $F$-pure modules over algebras of p^{-e}-linear operators. This shift in the viewpoint leads to a…

The aim of this paper is to study some distinguished classes of $k$-ideals of semirings, which include $k$-prime, $k$-semiprime, $k$-radical, $k$-irreducible, and $k$-strongly irreducible ideals. We discuss some of the properties of…

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via…

For any semiring, the concept of k-congruences is introduced, criteria for k-congruences are established, it is proved that there is an inclusion-preserving bijection between k-congruences and k-ideals, and an equivalent condition for the…

Tight closure test ideals have been central to the classification of singularities in rings of characteristic $p>0$, and via reduction to characteristic $p$, in equal characteristic zero as well. A summary of their properties and…

Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

Let f_1,...,f_r be homogeneous polynomials in K[x_1,...,x_n], K a field. Put F=y_1f_1+...+y_rf_r in K[x,y] and let I be the ideal of K[x,y] generated by the partials of F relative to the x_i and y_j. The Jacobian ring of F is the quotient…

We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed…

We prove that the category of commutative Hopf algebras over a field $k$ is co-semi-abelian. Consequently, the category of affine group $k$-schemes is semi-abelian. We establish coregularity by identifying the orthogonal factorization…

For any integer $d \geq 1$, we verify the Jacobian Conjecture for a $d$-linear map in two variables. We prove that almost all the coefficients of the formal inverse are in the ideal specified by the Jacobian condition. We find expressions…

We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties.…

The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from…

Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained…

We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial p_x with integer…

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly…

When $k$ is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated $k$-algebra as the closed subset of an ideal generated by appropriate minors of the so-called Jacobian matrix.…