Related papers: Distributive Noetherian Centrally Essential Rings

Let $G$ be a group with neutral element $e$ and let $S=\bigoplus_{g \in G}S_g$ be a $G$-graded ring. A necessary condition for $S$ to be noetherian is that the principal component $S_e$ is noetherian. The following partial converse is…

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a…

We consider a circle of ideas involving differential algebra, local Noetherian rings, and their generic formal fibers. Connecting these ideas gives rise to what we term a "twisted" subring $R$ of a ring $S$. Each such subring $R$ arises as…

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of…

We define the notion of central orderings for a general commutative ring $A$ which generalizes the notion of central points of irreducible real algebraic varieties. We study a central and a precentral loci which both live in the real…

We begin by investigating the class of commutative unital rings in which no two distinct elements divide the same elements. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and it equals…

Let $A$ be a ring and $\varphi$ its automorphism. It is proved that skew Laurent series ring $A((x,\varphi ))$ is a right serial ring if and only if $A$ is a right serial right Artinian ring.

In this short note we study the links of certain prime ideals of a noetherian ring R. We first give the definition of a link krull symmetric noetherian ring R. We then prove theorem 9 that states that for any linked prime ideals P' and Q'…

A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a strongly Z-graded ring is left quasi-duo if and only if it is right quasi-duo. This gives a partial answer to a problem posed by Dugas and Lam…

Let $R$ be a commutative ring with unity and let $X$ be an indeterminate over $R$. The \textit{Anderson ring} of $R$ is defined as the quotient ring of the polynomial ring $R[X]$ by the set of polynomials that evaluate to $1$ at $0$.…

In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this…

Let R be a ring with derivation d, such that (d(xy))^n =(d(x))^n(d(y))^n for all x,y in R and n>1 is a fixed integer. In this paper, we show that if R is a prime, then d = 0 or R is a commutative. If R is a semiprime, then d maps R in to…

We introduce the notion of an oscillatory formal distribution supported at a point. We prove that a formal distribution is given by a formal oscillatory integral if and only if it is an oscillatory distribution that has a certain…

We establish a characterization of dualizing modules among semidualizing modules. Let R be a finite dimensional commutative Noetherian ring with identity and C a semidualizing R-module. We show that C is a dualizing R-module if and only if…

Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra, and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented…

In this note we show that if a noetherian ring R is left and right Krull-homogenous and if: \Lambda ={P\textexclamdown {\epsilon} spec.R/ |R/P\textexclamdown|_r =|R|_r} and v ={Qj {\epsilon} spec.R| |R/Qj|l=|R|l} and P =\cap…

Let $(A,\mathfrak{m}, k=A/\mathfrak{m})$ be a noetherian local ring. Then it is equivalent $n = \dim A = \dim_k \mathfrak{m}/\mathfrak{m}^2$ and $\mathrm{Tor}^A_i(k,k) = 0$ for all $i \gg 0$. The article gives a proof with the…

The following representation theorem is proven: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean…

The celebrated Wedderburn-Artin theorem states that a simple left artinian ring is isomorphic to the ring of matrices over a division ring. We give a short and self-contained proof which avoids the use of modules.

The structure and the existence of maximal subrings in division rings are investigated. We see that if $R$ is a maximal subring of a division ring $D$ with center $F$ and $N(R)\neq U(R)\cup \{0\}$, where $N(R)$ is the normalizer of $R$ in…