Left Self-injective Ring Research Articles

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 ℵ0-Self-Injective Rings Modulo Their Jacobson Radical#

A well-known result by Y. Utumi states: Every right or left self-injective ring is von Neumann regular modulo its Jacobson radical. In this note, we give an example of a commutative ℵ0-self-injective ring which is not von Neumann regular modulo its Jacobson radical.

A Look at the Faith Conjecture

A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring then R is quasi-Frobenius. It was subse- quently conjectured by Carl Faith that every left (or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved the conjecture in the aArmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring with JORU 3 a 0: In this paper we construct a local ring R with JORU 3 a 0 and characterize when R is artinian or self- injective in terms of conditions on a bilinear mapping from a D-D-bimodule to D, where D is isomorphic to R=JORU. Our work shows that finding a counterexample to the Faith conjecture depends on the existence of a D-D-bimodule over a division ring D satisfying certain topological conditions.

SOME REMARKS ON THE FINITENESS CONDITIONS OF RINGS


 
 
 The aim of this paper is to study the finiteness of rings. We prove that if $A$ is a regular left-self-injective ring, then $A$ is of type III (purely infinite) implies that $E(A[x])$ is, and $A$ contains an abelian idempotent if and only if $E(A[x])$ contains an abelian idempotent. Also we prove that. 
 If $A$ is a regular left self-injective ring and $J$ is a left ideal in $A[x]$ such that $C(J)$ is an essential left ideal in $A$, then there exists a countably generated left ideal $J'$ in $A[x]$ such that $C(J')$ is an essential left ideal in $A$, and if $J'$ is an essential left ideal in $A[x]$, then $J$ is an essential left ideal in $A[x]$. 
 
 

Homological dimension and cardinality

Let {e(i) I i E f} be an infinite set of commuting idempotents in a ring R with 1 such that n m 1l e(ia) lJ (1-e(ia)) # 0 a=O 13=n+l for {ia I 0 M,,, then any right and left self-injective ring which is not semiperfect, any ring containing an infinite direct product of subrings, any ring containing the endomorphism ring of a countable direct sum of modules, and many quotient rings of such rings must all have infinite global dimension. This paper continues the investigation of the relationship between homological dimension and cardinality questions started in [4] and [5], combining the techniques of ?7 of [5] with a modification of a projective resolution in Pierce [7]. Employing a result of Hausdorff and Tarski, we show that the X corresponding to 2To plays an important role in the global dimension of rings where one can find analogues of characteristic functions of subsets of a set of orthogonal idempotents. 1. Homological dimension of an ideal generated by commuting idempotents. We first calculate the homological dimension of a right ideal of a ring R (with 1) generated by a set of idempotents, and then show that several types of rings possess such idempotents. A family t = {e(i) I i E f} of idempotents of R is called nice if (i) e(i)e(j) = e(j)e(i) Vi, j Ef . (ii) Ila= 1 e(ia) 171 n +1 (1-e(i))#O if {ia 1 R c R_n io < i < *-in Received by the editors December 1, 1969. AMS Subject Classifications. Primary 1690; Secondary 0430, 1640.