Related papers: Graded Perinormality

In this study we consider domains that are composed of an infinite sequence of self-similar rings and corresponding finite element spaces over those domains. The rings are parameterized using piecewise polynomial or tensor-product B-spline…

Let $R$ be a commutative $G$-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal $I$ of $R$ is said to be graded radically principal if $Grad(I)=Grad(\langle…

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $I_{e}$-prime submodule as a generalization of a graded prime…

Semi-standard graded rings are a generalized notion of standard graded rings. In this paper, we compare generalized notions of the Gorenstein property in semi-standard graded rings. We discuss the commonalities between standard graded rings…

The goal of this article is to present the graded weakly $S$-primary ideals and $g$-weakly $S$-primary ideals which are extensions of graded weakly primary ideals. Let $R$ be a commutative graded ring, $S\subseteq h(R)$ and $P$ be a graded…

This paper studies the average gradient over the local region of a function and constructs the homogenization function of a function. It is found that there are some good properties about the local extreme points and the global extreme…

Estimation of a multivariate regression function from independent and identically distributed data is considered. An estimate is defined which fits a deep neural network consisting of a large number of fully connected neural networks, which…

We introduce a notion of generalized homogeneous derivations on graded rings as a natural extension of the homogeneous derivations defined by Kanunnikov. We then define gr-generalized derivations, which preserve the degrees of homogeneous…

We observe algebraic derivations on an affine domain B defined over an algebraically closed field of characteristic 0, which are called locally finite derivations in commutative and non-commutative contexts in other references. We observe…

Let $M$ be G-graded R-module. The idea of a graded weakly primal submodule of $M$, which is a generalization of a graded primal submodule, is introduced and discussed in this paper. Some characteristics and characterizations are assigned to…

An integral domain $D$ is called a \emph{prime-divisor-finite domain} (PDF-domain) if every nonzero element has only finitely many nonassociate prime divisors. A domain $D$ is said to be a \emph{tightly prime-divisor-finite domain}…

We study the "local" behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We deal with the "global" problem of building a new semistar operation on a given integral…

A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but…

We consider nonlocal equations of order larger than one with measure data and prove gradient regularity in Sobolev and H\"older spaces as well as pointwise bounds of the gradient in terms of Riesz potentials, leading to fine regularity…

Let $\Gamma$ be a cancellation monoid and $R=\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded ring. It is shown that $R$ is graded left semihereditary if and only if $R$ is graded left coherent and every graded submodule of a…

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain graded by an arbitrary grading torsionless monoid $\Gamma$, and $\star$ be a semistar operation on $R$. In this paper we define and study the graded integral domain…

Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $\phi: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper…

Call a domain $R$ an sQQR-domain if each simple overring of $R$, i.e., each ring of the form $R[u]$ with $u$ in the quotient field of $R$, is an intersection of localizations of $R$. We characterize Pr\"ufer domains as integrally closed…

For a graded domain $R=k[X_0,...,X_m]/J$ over an arbitrary domain $k$, it is shown that the ideal generated by elements of degree $\geq mA$, where $A$ is the least common multiple of the weights of the $X_i$, is a normal ideal.

The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$…