A semiperfect one-sided injective ring

We construct an example of a ring R: such that i) R is semiperfect, ii) R is right but not left self-injective, iii) R is an essential extension of its socle on the left but not on the right.     

Von neumaivn regularity of sf-rings

A ring R is called left (right) SF-ring if all simple left (right) R-modules are flat. It is proved that R is Von Neumann regular if R is a right SF-ring whoe maximal essential right ideals are ideals. This gives the positive answer to a qestion proposed by R. Yue Chi MIng in 1985, and a counterexample is given to settle the follwoing question in the negative: If R is an ERT ring which is one-sided V-ring, is R a left and right V-ring? Some other conditions are given for a SF-ring to be regular.     

Self-injective rings and endomorphisms of free modules

It is shown that the ring of endomorphisms of an arbitrary free R-module is right self-injective if and only if R is quasi-Frobenius.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 533–540, November, 1969.We take this opportunity to thank L. A. Skornyakov for his guidance.     

The Relative Properties of Graded Ring R and Smash Product R # G

ＴｈｅＲｅｌａｔｉｖｅＰｒｏｐｅｒｔｉｅｓｏｆＧｒａｄｅｄＲｉｎｇＲａｎｄ　ＳｍａｓｈＰｒｏｄｕｃｔＲ＃ＧＷｅｉＪｕｎｃｈａｏ（魏俊潮）；ＬｉＬｉｂｉｎ（李立斌）（ＹａｎｇｚｈｏｕＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，Ｙａｎｇｚｈｏｕ，２２５０...     

Semiprime rings with bounded indices of nilpotent elements

This paper deals with the structure of semiprime rings for which the indices of the nilpotent elements are bounded. It is shown that the complete right ring of quotients of such a ring is a regular, right self-injective ring in which each finitely generated ideal is generated by a central idempotent. The indices of the nilpotent elements of the factor ring of such a ring with respect to a minimal prime ideal do not exceed the upper bound of the indices of the nilpotent elements of the original ring. A criterion for the regularity (in the sense of von Neumann) of such rings is obtained. Also investigated are right completely idempotent rings with bounded indices of nilpotent elements (it is shown, in particular, that each nonzero ideal of such a ring contains a nonzero central idempotent).Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 237–249, 1988.     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

Distributive and Semihereditary Rings

Let A be a right and left distributive ring. For a positive integer n, we obtain a criterion of projectivity of all n-generated right ideals of the ring A and a criterion of the right semi-heredity of the ring A.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 253–258, 2003.     

自内射环的余Hopf性

本文证明了自内射环R是余Hopf的当且仅当R满足stablerangeone.于是得到了Varadarajan在[9]中的公开总是对于自内射环是成立的,即M_n(R)是余Hopf的当且仅当R是余Hopf的.作为应用证明了Goodeal的一个公开问题对于自内射正则环有肯定的回答.     

Rings over which all modules are <Emphasis Type="Italic">I</Emphasis><Subscript>0</Subscript>-modules

Let A be a ring that does not contain an infinite set of idempotents that are orthogonal modulo the ideal SI(A _A ). It is proved that all A-modules are I ₀-modules if and only if either A is a right semi-Artinian, right V-ring or A/SI(A _A ) is an Artinian serial ring and the square of the Jacobson radical of A/SI(A _A ) isequal to zero. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 193–200, 2007.     

A note on quasi-Frobenius extensions

It is shown that for a quasi-Frobenius extension A of a right non-singular ring B if A is a right self-injective ring, then so is B.     

Elementary Reduction of Matrices over Right 2-Euclidean Rings

We introduce a concept of noncommutative (right) 2-Euclidean ring. We prove that a 2-Euclidean ring is a right Hermite ring, a right Bezout ring, and a GE _n -ring. It is shown that an arbitrary right unimodular string of length not less than 3 over a right Bezout ring of stable rank possesses an elementary diagonal reduction. We prove that a right Bezout ring of stable rank 1 is a right 2-Euclidean ring.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1717 – 1721, December, 2004.     

Recent development of Faith conjecture

Is a semiprimary right self-injective ring a quasi-Frobenius ring? Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF: a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.     

On a class of non-noetherian V-ring

Right V-rings R with infinitely generated right socle SOC(R_R) such that R/SOC(R_R) is a division ring are characterized as those non-noetherian rings over which a cyclic right module is either non-singular or injective. Furthermore, it is shown that a non-noetherian, right V-ring S is Morita-equivalent to a ring of this type iff all singular simple right S-modules are isomorphic and every direct sum of uniform modules with an injective module over S is extending.     

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 ?₀-self-injective ring which is not von Neumann regular modulo its Jacobson radical.     

Dedekind Finite Rings and a Theorem of Kaplansky

Abstract

A ring Ris Dedekind Finite(=DF) if xy = 1 implies yx = 1 for all x, yin R. Obviously any subring of a DFring Ris DF. The object of the paper is to generalize, and give a radically new proof of a theorem of Kaplansky on group algebras that are Dedekind finite. We shall prove that all right subrings of right and left self-injective (in fact, continuous) rings are DF.     

On Decompositions of Quasi-Regular Rings

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Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

Description of Barely Transitive Groups with Soluble Point Stabilizer

We consider a ring whose left modules of finite length are semisimple and prove some results on such a ring. We also consider when such a ring is a left V-ring.     

The right ideals of an alternative ring

It is proved that if P is a right ideal and I a two-sided ideal of an alternative ring A, then neither P² nor IP is in general a right ideal of A. Moreover, it is shown that in the alternative ring A the right annihilator of the right ideal P, i.e., the setE _r(P) = { A|Pz = 0}, is not necessarily either a left or a right ideal, nor even a subring of A.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 239–242, September, 1972.