Mohamed F. Yousif

Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ⊆ess M means that N is an essential submodule of M.

WEAKLY CONTINUOUS AND C2-RINGS

A ring R is called right weakly continuous if the right annihilator of each element is essential in a summand of R, and R satisfies the right C2-condition (every right ideal that is isomorphic to a direct summand of R is itself a direct summand). We show that a ring R is right weakly continuous if and only if it is semiregular and J(R) = Z(R R ). Unlike right continuous rings, these right weakly continuous rings form a Morita invariant class. The rings satisfying the right C2-condition are studied and used to investigate two conjectures about strongly right Johns rings and right FGF-rings and their relation to quasi-Frobenius rings.

ON P-INJECTIVE RINGS

1. Definitions and preliminary results. Throughout this paper R will be an associative ring with unity and all /?-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X) (resp. \(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z{RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A^M, the notation A c® M will mean that A is a direct summand of M. A module MR is called p-injective if for every a e R, every 7?-linear map from aR to A/ can be extended to an ^-linear map from R to M. R is called right p-injective if RR is p-injective. Recall that a module MR is called uniserial if its submodules are linearly ordered by inclusion and serial if it is a direct sum of uniserial submodules. A ring R is right uniserial (serial) if RR is uniserial (serial). We record some well-known results on serial and p-injective rings.

On perfect simple-injective rings

Harada calls a ring R right simple-injective if every R-homomorphism with simple image from a right ideal of R to R is given by left multiplication by an element of R. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if R is left perfect and right simple-injective, then R is quasi-Frobenius if and only if the second socle of R is countably generated as a left R-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings. A ring R is called quasi-Frobenius if R is left (and right) artinian and left (and right) self-injective. A well known result of Osofsky [15] asserts that a left perfect, left and right self-injective ring is quasi-Frobenius. It has been conjectured by Faith [9] that a left (or right) perfect, right self-injective ring is quasi-Frobenius. This conjecture remains open even for semiprimary rings. Throughout this paper all rings R considered are associative with unity and all modules are unitary R-modules. We write MR to indicate a right R-module. The socle of a module is denoted by soc(M). We write N ⊆ M (N ⊆ess M) to mean that N is a submodule (essential) of M . For any subset X of R, l(X) and r(X) denote, respectively, the left and right annihilators of X in R. A ring R is called right Kasch if every simple right R-module is isomorphic to a minimal right ideal of R. The ring R is called right pseudo-Frobenius (a right PF-ring) if RR is an injective cogenerator in mod-R; equivalently if R is semiperfect, right self-injective and has an essential right socle. A ring R is called right principally injective if every R-morphism from a principal right ideal of R into R is given by left multiplication. In [14], a ring R is called a right generalized pseudo-Frobenius ring (a right GPF-ring) if R is semiperfect, right principally injective and has an essential right socle. We write J = J(R) for the Jacobson radical of the ring R. Following Fuller [10], if R is semiperfect with a basic set E of primitive idempotents, and if e, f ∈ E, we say that the pair (eR,Rf) is an i-pair if soc(eR) ∼= fR/fJ and soc(Rf) ∼= Re/Je. Received by the editors April 24, 1995 and, in revised form, October 11, 1995. 1991 Mathematics Subject Classification. Primary 16D50, 16L30.

Soc-Injective Rings and Modules*

ABSTRACT If M and N are right R-modules, M is called Socle-N-injective (Soc-N-injective) if every R-homomorphism from the socle of N into M extends to N. Equivalently, for every semisimple submodule K of N, any R-homomorphism f : K → M extends to N. In this article, we investigate the notion of soc-injectivity.

Annihilators and the CS-condition

It is proved that if every cyclic right /^-module is torsionless and R is a left CS-ring then R is semiperfect left continuous with soc(RR) essential in RR. As a consequence every right cogenerator, left CS-ring R is shown to be right pseudo-Frobenius and left continuous, and an example is given to show that R need not be left selfinjective. It is also proved that if R is a left CS-ring and every cyclic right /^-module embeds in a free module, then R is quasi-Frobenius if and only if J(R) c Z(RR).

On Quasi-Frobenius Rings

There are three outstanding conjectures about quasi-Frobenius rings: The Faith conjecture that every left perfect, right selfinjective ring is quasi-Frobenius; The FGF-conjecture that every ring for which each finitely generated right module embeds in a free module is quasi-Frobenius; and The Faith-Menal conjecture that every right noetherian ring in which all right ideals are annihilators is quasi-Frobenius. In this paper we survey recent work on these conjectures and provide some new results on the subject.

Continuous rings with chain conditions

The quasi-Frobenius rings are characterized as the left continuous rings satisfying either (A,) or (Al) and either (S,) or (S,), where these conditions are defined as follows: (A,): ACC on left annihilators; (AZ): R/X (S,): S = r(/(S)) for every minimal right ideal S; and (S,): Every minimal right ideal is essential in a summand of RR. These characterizations extend several results in the literature. In addition, it is shown that, in these rings, Soc(RR) = Soc( RR), Soc(eR) is simple for every primitive idempotent e of R. and there exists a complete set of distinct representatives [Rt,, , Rtn} of the isomorphism classes of the simple left R-modules such that (t, R, , t,R) is a complete set of distinct representatives of the isomorphism classes of the simple right R-modules.

FP-Injective, Simple-Injective, and Quasi-Frobenius Rings

Abstract Motivated by a result of Mochizuki [Mochizuki, H. (1965). Finitistic global dimension for rings. Pacific J. Math.15(1):249–258] that, for a left perfect ring R with Jacobson radical if and only if R is right self-injective, we prove here that, for a semiperfect ring R with essential right socle S r , (1) R is right FP-injective if every R-homomorphism from a finitely generated small submodule of a free right R-module F to S r can be extended to an R-homomorphism from F to R, (2) R is right simple-injective if every R-homomorphism from a small right ideal of R to R with simple image can be extended to an R-homomorphism from R to R, and (3) R is right self-injective if every R-homomorphism from a small right ideal of R to R can be extended to an R-homomorphism from R to R. As consequences, several known results on right self-injective rings, right FP-injective rings and right simple-injective rings are extended to larger classes of rings.

Extensions of simple-injective rings

A ring Ris called right simple-injective if every itMinear map with simple image from a right ideal to Rcan be extended to R. We characterize when matrix rings, upper triangular matrix rings and trivial extensions are right simple-injective. We also study split null extensions of simple-injective rings.