Related papers: Herzog ideals and $F$-singularities

We present an overview of the close analogies between the character rings of finite groups and the fusion rings of rational conformal models, which follow from general principles related to orbifold deconstruction.

A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active…

The notion of clean rings and 2-good rings have many variations, and have been widely studied. We provide a few results about two new variations of these concepts and discuss the theory that ties these variations to objects and properties…

Let R be a differential domain finitely generated over a differential field, F, with field of constants, C, of characteristic 0. Let E be the quotient field of R. The paper investigates necessary and sufficient conditions on R's…

We prove that all finite W-algebras associated with nilpotent elements e in a complex semisimple Lie algebra g have finite-dimensional representations. In order to obtain this result we establish a connection between primitive ideals of…

In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal of…

Let $\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gr\"obner cone $\mathcal{C}_{\mathcal{A}}$ parametrizing term orders inducing an initial degeneration of the ideal $I_{\mathcal{A}}$ of relations among the cluster…

For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine map alpha:Y -> G (where Y is an element of the open coset ring of H) induces a…

Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over…

Many Hilbert modules over the polynomial ring in m variables are essentially reductive, that is, have commutators which are compact. Arveson has raised the question of whether the closure of homogeneous ideals inherit this property and…

Let $G$ be a locally profinite group and let $k$ be a field of positive characteristic $p$. Let $Z(G)$ denote the center of $G$ and let $\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra of natural endomorphisms…

Let $R$ be a commutative ring with nonzero identity. A. Yassine et al. defined in the paper (Yassine, Nikmehr and Nikandish, 2020), the concept of $1$-absorbing prime ideals as follows: a proper ideal $I$ of $R$ is said to be a…

The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard…

The a-invariant, the F-pure threshold, and the diagonal F-threshold are three important invariants of a graded K-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly F-regular rings. In this…

Let $f : A \rightarrow B$ be a ring homomorphism and $J$ be an ideal of $B$. In this paper, we give a characterization of zero divisors of the amalgamation which is a generalization of Maimani's and Yassemi's work (see \cite{y}). Also, we…

In 1999 Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal $I$ is componentwise linear if for all non-negative integers $d$, the ideal generated by the homogeneous elements of degree $d$ in $I$ has a linear…

The purpose of this work is to extend the study of the commutative rings whose lattice of ideals can be a structure of BL-algebra as carry out by Heubo et al in 2018, to non commutative rings appointed in the work as pseudo BL-rings. We…

We study the algebraic sets of pairs of matrices defined by the vanishing of the anti-diagonal as well as the cross-diagonal of their commutator matrix. We prove that, over a field of prime characterisitic, the coordinate ring of the latter…

In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.

It is known that every nonzero Jordan ideal of $2$-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero ideal. This can be…