Dinh Van Huynh

WHEN SELF-INJECTIVE RINGS ARE QF: A REPORT ON A PROBLEM

Theorems of Osofsky and Kato imply that a right and left self-injective one-sided perfect ring is quasi-Frobenius (= QF). The corresponding question for one-sided self-injective one or two-sided perfect rings remains open, even assuming that the ring is semiprimary. The latter version of the problem is known as Faiths Conjecture (FC). We survey results on QF rings, especially those obtained in connection with FC. We also review various results that provide partial answers to another problem of Faith: Is a right FGF ring necessarily QF? On this topic, we provide a new result, namely that if all factor rings of R are right FGF, then R is QF (Theorem 6.1). In Sec. 7 we review results concerning the question of when a D-ring is QF. Sections 8 and 9 are devoted respectively to IF rings, and to Σ-injective rings and Σ-CS rings.

A characterization of artinian rings

Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R -module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R -module is finitely embedded.

Structure of some noetherian SI rings

Abstract We describe the structure of rings over which every cyclic (or finitely generated) right module is a direct sum of a projective module and an injective module.

Rings characterized by direct sums of cs modules

It is shown that a ring for which every CS right module is ∑CS is right artinian. As a consequence, it is also shown that over a ring R every direct sum of CS right R-modules is CS iff R is right artinian and the composition length of every uniform right R-module is at most 2.

A characterization of rings with Krull dimension

Throughout this paper rings will mean associative rings with identity and all modules are assumed to be unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, see for example [ 1, 2, 9-111. One of the most important results in this direction is the result of Osofsky [S, Theorem] which says that a ring R is semisimple (i.e., R is right Artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. Starting from this and in connection with a result of Vamos [ 1 l] the following result has been recently obtained in [2, Theorem 1.11: A ring R is right Artinian if and only if every cyclic right R-module is a direct sum of an injective module and a finitely cogenerated module. In the present paper we follow this investigation and aim to show a similar result for rings with Krull dimension. We consider the following condition about a ring R:

Some Results on Self-Injective Rings and Σ-CS Rings

Abstract A module M is CS if every submodule of M is essential in a direct summand of M. In this note we use the CS condition to provide conditions for semiperfect rings to be self-injective. Further we show that every finitely generated CS right module over a right semi-artinian ring has finite uniform dimension. Using this, we prove that if R is a right semi-artinian ring such that is CS, then is also CS for any set A. Moreover, R is then right and left artinian.

On the symmetry of the Goldie and CS conditions for prime rings

It is shown that: (a) If R is a prime right Goldie right CS ring with right uniform dimension at least 2, then R is left Goldie, left CS; (b) A semiprime ring R is right Goldie left CS iff R is left Goldie, right CS. All rings are associative having an identity and all modules are unitary. A right module M over a ring R is called CS (or extending) if every submodule of M is essential in a direct summand of M , or equivalently, if every complement submodule of M is a direct summand of M . A ring R is called right CS (resp., left CS), if RR (resp., RR) is a CS module. CS modules have been extensively studied by many authors. A ring R is defined to be a right (left) Goldie ring if R has ascending chain condition on right (left) annihilators and the right (left) uniform dimension of R is finite. A right Goldie ring R is (semi-)prime if and only if R has classical right quotient ring which is (semi-)simple artinian. For notation not defined here we refer the reader to [1], [2] and [3]. Theorem 1. A prime right Goldie, right CS ring R with right uniform dimension at least 2, is left Goldie, and left CS. Proof. Let n be the right uniform dimension of R. By assumption, n ≥ 2. Since R is right CS, R = e1R ⊕ · · · ⊕ enR where each eiR is uniform and {ei}i=1 is a system of orthogonal idempotents of R. Let Q be the classical right quotient ring of R. Then we have: R ∼=  e1Re1 e1Re2 · · · e1Ren e2Re1 e2Re2 · · · e2Ren . . · · · . . . · · · . . . · · · . enRe1 enRe2 · · · enRen  ⊆  e1Qe1 e1Qe2 · · · e1Qen e2Qe1 e2Qe2 · · · e2Qen . . · · · . . . · · · . . . · · · . enQe1 enQe2 · · · enQen  ∼= Q. Received by the editors May 12, 1998 and, in revised form, September 28, 1998 and December 9, 1998. 1991 Mathematics Subject Classification. Primary 16P60, 16N60, 16D80. c ©2000 American Mathematical Society

On a class of non-noetherian V-ring

Right V-rings R with infinitely generated right socle SOC(RR) such that R/SOC(RR) 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.

AN APPROACH TO BOYLE'S CONJECTURE

A ring R is called a right Ql-ring if every quasi-injective right K-module is injective. The well-known Boyles Conjecture states that any right Ql-ring is right hereditary. In this paper we show that if every continuous right module over a ring R is injective, then R is semisimple artinian. In fact, if every singular continuous right R-module satisfying the restricted semisimple condition is injective, then R is right hereditary. Moreover, in this case, every singular right R-module is injective.

Rings whose finitely generated modules are extending

A module M is called an extending (or CS) module, if every submodule of M is essential in a direct summand of M. In this paper we show that a ring R is right noetherian if every finitely (or 2-) generated right R-module is extending.