Related papers: On valuation rings

A submodule $W$ of a p-primary module $M$ of bounded order is known to be regular if $W$ and $M$ have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

Let $A$ be a ring of dimension $d$ containing an infinite field $k$, $T_1,\ldots,T_r$ be variables over $A$ and $P$ be a projective $A[T_1,\ldots,T_r]$-module of rank $n$. Assume one of the following conditions hold. (1) $2n\geq d+3$ and…

If $E$ is a subset of the integers then the $n$-th characteristic ideal of $E$ is the fractional ideal of $\mathbb{Z} $ consisting of $0$ and the leading coefficients of the polynomials in $\mathbb{Q}[x]$ of degree no more than $n$ which…

Let (R,m) be a noetherian local ring and let $\mathcal{C}$ be the class of all R-modules M which possess a reflexive submodule U such that M/U is finitely generated. For every R-module $M\in \mathcal{C}$ the canonical embedding $\varphi:…

The aim of this paper is to introduce and study graded and filtered gamma rings and gamma modules. We prove that the filtered $\Gamma$-ring (module) is a generalization of the notion of graded ring (module). Also, we construct a graded…

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the…

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM…

Let R be a ring (associative, with 1). A non-zero module M is said to be a Pruefer module provided there exists a surjective, locally nilpotent endomorphism with kernel of finite length. The aim of this note is construct Pruefer modules…

Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These…

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component,…

We give a criterion for rings with $\m^3=0$ which are obtained as connected sums of two other rings to have non-trivial totally acyclic modules.

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is…

In this note we investigate the question whether a henselian valued field carries a non-trivial 0-definable henselian valuation (in the language of rings). It follows from the work of Prestel and Ziegler that there are henselian valued…

This paper presents an extension of the concept of NR-clean introduced in [12] to graded ring theory. We define and explore graded NR-clean rings, which generalize the class of graded U-nil clean previously studied in [15]. We provide…

Let $G$ be a group with identity $e$ and $R$ a commutative $G$-graded ring with a nonzero unity $1$. In this article, we introduce the concepts of graded $r$-submodules and graded special $r$-submodules, which are generalizations for the…

Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit…

In this article, we consider the structure of graded rings, not necessarily commutative nor with unity, and study the graded weakly prime ideals. We investigate the graded rings in which all graded ideals are graded weakly prime. Several…