Restricted Minimum Condition Research Articles

Restricted Minimum Condition in Reduced Commutative Rings

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, the factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains the RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. In the second part, we will study CDR domains, i.e., domains where for any two ideals I, J the inclusion \(I\subseteq J\) implies that I is a multiple of J. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials R[x] is RM if and only if R is a reduced Artinian ring. And we will study the relation between RM rings and UFDs.

Restricted minimum condition for group-rings and matrix extensions

A ring R is said to have right restricted minimum condition ( for short) if R/A is an Artinian right R-module for any essential right ideal A of R. We study the for matrix extensions of a ring R and for the group-ring RG. Among other things, it is proven that having r.RMC is a Morita invariant property for rings. For the upper triangular n by n matrix ring over R has r.RMC if and only if RR is Artinian. If G is a not torsion abelian group then RG has r.RMC if and only if R is a semisimple ring and where H is a finite group whose order is invertible in R. If X is a completely regular topological space, then the ring C(X) has if and only if X is finite. Many examples of rings with r.RMC are presented.

On the Restricted Minimum Condition for Rings

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition ( $${\mathrm{r.RMC}}$$ , for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with $${\mathrm{r.RMC}}$$ quasi-Frobenius? (ii) Whether a serial ring with $${\mathrm{r.RMC}}$$ must be Noetherian? We carry out a study of rings with $${\mathrm{r.RMC}}$$ and determine when a right extending ring has $${\mathrm{r.RMC}}$$ in terms of rings $${\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}$$ such that S is right Artinian, $$M_{Q}$$ is semisimple ( $$Q={\mathrm{Q}}(R)$$ ) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with $${\mathrm{r.RMC}}$$ is quasi-Frobenius if and only if $$\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)$$ if and only if $$\hbox {Z}_{r}(R)$$ is a finitely generated left ideal and $${\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})$$ is a finitely generated right ideal. Right serial rings with $${\mathrm{r.RMC}}$$ are studied and proved that a non-singular serial ring has $${\mathrm{r.RMC}}$$ if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that $$\mathrm{RMC}$$ is not symmetric for a ring.

On modules and rings with the restricted minimum condition

A module $M$ satisfies the restricted minimum condition if $M/N$ is artinian for every essential submodule $N$ of $M$. A ring $R$ is called a right RM-ring whenever $R_R$ satisfies the restricted minimum condition as a right module. We give several struct

A NOTE ON SELF-SMALL MODULES OVER RM-DOMAINS

This paper provides two new characterizations of RM-domains, i.e. domains satisfying the restricted minimum condition. A Noetherian domain is a RM-domain if and only if every torsion module is a direct sum of submodules whose cyclic submodules have finite length and homogeneous composition series. We show that this occurs exactly if all self-small torsion modules are finitely generated.

A ONE-SIDED PRIME IDEAL PRINCIPLE FOR NONCOMMUTATIVE RINGS

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.

On Serial Noetherian Rings

It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian.

On rings with restricted minimum condition

On rings with restricted minimum condition

Rings with Enough Invertible Ideals

All rings are associative with identity element 1 and all modules are unital. A ring has enough invertible ideals if every ideal containing a regular element contains an invertible ideal. Lenagan [8, Theorem 3.3] has shown that right bounded hereditary Noetherian prime rings have enough invertible ideals. The proof is quite ingenious and involves the theory of cycles developed by Eisenbud and Robson in [5] and a theorem which shows that any ring S such that R ⊆ S ⊆ Q satisfies the right restricted minimum condition, where Q is the classical quotient ring of R. In Section 1 we give an elementary proof of Lenagan's theorem based on another result of Eisenbud and Robson, namely every ideal of a hereditary Noetherian prime ring can be expressed as the product of an invertible ideal and an eventually idempotent ideal (see [5, Theorem 4.2]). We also take the opportunity to weaken the conditions on the ring R.

The Restricted Minimum Condition in Noetherian Hereditary Rings

The Restricted Minimum Condition in Noetherian Hereditary Rings

The Width of a Module

An R-module N is said to have finite width n if n is the smallest integer such that for any set of n + 1 elements of N, at least one of the elements is in the submodule generated by the remaining n. The width of N over R will be denoted by W(R, N).The notion of width was introduced by Brameret [2, p. 3605]. However, Cohen [3] investigated rings of finite rank, which, in the case that R is a local Noetherian domain, is equivalent to width (Proposition 1.6). He showed that finite width of R was both equivalent to R having Krull dimension one, and to R having the restricted minimum condition (Theorem 1.12).

Rings with Restricted Minimum Condition

Rings with Restricted Minimum Condition

Rings with restricted minimum condition

Definition. A ring R is said to satisfy the restricted minimum condition (or to be a RM ring, for short), if for each ideal A $ (0) in R, the ring R/A is right artinian. In this paper we consider a RM ring, and are furthermore interested in the case where R itself is not right artinian. The concept of a commutative RM ring was introduced by I. S. Cohen [1], who also proved the following results for a commutative ring R: (a) R is RM iff R is noetherian and every proper prime ideal is maximal. (b) R is RM but not artinian iff R is a noetherian integral domain not a field, in which every proper prime ideal is maximal.

Commutative rings with restricted minimum condition

Commutative rings with restricted minimum condition