Related papers: Herzog ideals and $F$-singularities

The F-thresholds are characteristic p analogs of the jumping coefficients for multiplier ideals in characteristic zero. In this article we give an alternative description of the F-thresholds of an ideal in a regular and F--finite ring $R$.…

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the…

We exhibit a class of Hibi rings which are diagonally F-regular over fields of positive characteristic, and diagonally $F$-regular type over fields of characteristic zero, in the sense of Carvajal-Rojas and Smolkin. It follows that such…

We consider singular integrals associated to homogeneous kernels on self similar sets. Using ideas from ergodic theory we prove, among other things, that in Euclidean spaces the principal values of singular integrals associated to real…

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a…

In this paper, we introduce $\phi$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\neq0$ and $\phi:\mathcal{I}(R)\rightarrow\mathcal{I}(R)\cup\{\emptyset\}$ be a function where…

We propose a general quantum Hamiltonian formalism of a renormalization group (RG) flow with an emphasis on generalized symmetry by interpreting the elementary relationship between homomorphism, quotient ring, and projection. In our…

For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $\omega_R$ and the property that the socle…

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a…

(1) Let $(A,\mathfrak{m})$ be complete Noetherian local ring of dimension $d$ and let $P$ be a prime ideal with $G_P(A) = \bigoplus_{n \geq 0}P^n/P^{n+1}$ a domain. Fix $r \geq 1$. If $J$ is a homogeneous ideal of $G_{P^r}(A)$ with…

We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $O_X(*Z)f^{1-\alpha}$ associated to the effective $\mathbb{Q}$-divisor $D=\alpha\cdot Z$, where $0<\alpha\le1$ and $Z=(f=0)$ is an irreducible hypersurface…

For every $p\geq 2$, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to the positive linear functionals on the free group $C^*$-algebra associated with the ideal…

The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many…

In this paper for a noetherian ring R with nilradical N we define semiprime ideals P and Q called as the left and right krull homogenous parts of N . We also recall the known definitions of localisability and the weak ideal invariance…

For a reduced F-finite ring R of characteristic p >0 and q=p^e one can write R^{1/q} = R^{a_q} \oplus M_q, where M_q has no free direct summands over R. We investigate the structure of F-finite, F-pure rings R by studying how the numbers…

This work is motivated by the papers [EG85] and [Ngu15] in which the following two problems are solved. Let $\mathcal{O}$ is a finitely generated $\mathbb{Z}$-algebra that is an integrally closed domain of characteristic zero, consider the…

In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair $(R, \Delta)$. We call these analogues centers of $F$-purity. We prove positive characteristic analogues of subadjunction-like results,…

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This…

The homogeneous weights and the M\"obius functions and Euler phi-functions on finite rings are discussed; some computational formulas for these functions on finite principal ideal rings are characterized; for the residue rings of integers,…