Related papers: A Generalized Serre's Condition

Let $\mathcal{S}$ be an iterated function system in $\mathbb{R}^d$, with full support and some restrictions on the allowable rotations. We show that $\mathcal{S}$ satisfies the weak separation condition if and only if it satisfies the…

This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring $R$ of prime characteristic $p$. Let $R^{\circ}$ denote the complement in $R$ of the union of the minimal prime ideals of $R$. A big…

Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for…

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre's condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_p is…

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field %$k$ of characteristic $p > 0$. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context…

In 1994, Sule presented the necessary and sufficient conditions of the feedback stabilizability of systems over unique factorization domains in terms of elementary factors and in terms of reduced minors. Recently, Mori and Abe have…

Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only…

Let $R$ be a complete regular local ring with an algebraically closed residue field and let $A$ be a Noetherian $R$-subalgebra of the polynomial ring $R[X]$. It has been shown in \cite{DO2} that if $\dim R=1$, then $A$ is necessarily…

Let $R$ be a commutative Noetherian ring and $\alpha$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[\theta; \alpha]$ satisfy the property $(\diamond)$, that for every simple $S$-module…

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction…

Let $R$ be a commutative noetherian ring, and let $\mathscr{S}$(resp. $\mathscr{L}$) be a Serre(resp. localizing) subcategory of the category of $R$-modules. If $\Bbb F$ is an unbounded complex of $R$-modules Tor-perpendicular to…

In this paper we introduce the definition of a noetherian disjoint ring and that of a noetherian non-disjoint ring . For a noetherian ring R , with nilradical N if P and Q represent the semiprime ideals of R called as the right and the left…

We prove a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient…

Let $R$ be a commutative ring with identity and $S \subseteq R$ be a multiplicative set. An ideal $Q$ of $R$ (disjoint from $S$) is said to be $S$-primary if there exists an $s\in S$ such that for all $x,y\in R$ with $xy\in Q$, we have…

Let R be a semiperfect commutative Noetherian ring and C a semidualizing R-module. We study the theory of linkage for modules of finite G_C-dimension. For a horizontally linked R-module M of finite G_C-dimension, the connection of the Serre…

A cover of an associative (not necessarily commutative nor unital) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the…

In this paper we prove that if $R$ is a commutative refinement ring and $M$, $N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal $m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every…

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R…

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. In particular, for a subgroup H and a normal subgroup N of H,…