Rings in which every right ideal is quasi-injective

Jain, S. K.; Mohamed, Saad H.; Singh, Surjeet

doi:10.2140/pjm.1969.31.73

Cited by 39 publications

(34 citation statements)

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“…We conclude that A is simple Artinian and (2) implies (1). • Theorem 2.4 slightly extends a result of Jain, Mohamed and Singh [14]. All the results in this note are concerned in some way or other with generalizations of injective or projective modules.…”

Section: Scs Modulessupporting

confidence: 68%

A Generalization of Injectivity for Modules Over a Unitary Ring

Ming¹

2006

Demonstratio Mathematica

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Abstract. In this note, we consider certain generalizations of injectivity and pinjectivity in connection with von Neumann regular rings, self-injective regular rings, II-regular rings, semi-simple Artinian and simple Artinian rings.A generalization of quasi-injective modules, noted SCS modules, is introduced. It is proved that A is a left self injective regular ring if, and only if, A is a left p-injective left non-singular left SCS ring. SCS rings are used to characterize simple Artinian rings. A generalization of p-injective modules, noted WGP-injective is used to study II-regular rings. If A is a right p.p. right WGP-injective ring, then A is II-regular. If A is a semi-prime ring whose simple left modules are either WGP-injective or projective, then the centre of A is von Neumann regular. Left Artinian rings are characterized as left Noetherian rings whose prime factor rings are left WGP-injective. Also, A is a left WGP-injective ring if and only if for any a 6 A, there exists a positive integer n such that a n A is a right anihilator. (Here a n may be zero.) IntroductionThroughout, A denotes an associative ring with identity and A-modules are unitial, J, Y, Z will stand respectively for the Jacobson radical, the right singular ideal and the left singular ideal of A. A is called respectively semiprimitive ( (a) Following Faith [7], A is called VNR if 11 A is a von Neumann regular ring". Recall that:

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Section: Scs Modulessupporting

confidence: 68%

A Generalization of Injectivity for Modules Over a Unitary Ring

Ming¹

2006

Demonstratio Mathematica

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“…A ring R is called a q-ring if each right ideal of R is quasi-injective. The study of q-rings was initiated by Jain, Singh, and Mohamed (13). In (6) ; Caldwell called a ring R hypercyclic if each cyclic right i?-module has a cyclic injective hull.…”

Section: Preliminariesmentioning

confidence: 99%

On rings with quasi-injective cyclic modules

Javed

1974

Proceedings of the Edinburgh Mathematical Society

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A ring R is called a qc-ring if each cyclic R-module is quasi-injective. For various properties of these rings we refer to Ahsan (1) and Koehler (15). In this paper we shall obtain some additional results related to qc-rings. The scheme of the paper is as follows. Section 2 contains various preliminary definitions and results. In Section 3, we shall prove that every commutative hypercyclic ring is a qc-ring. In this section, we shall also show that a qc-ring which satisfies the ascending chain condition on its annihilators has nilpotent Jacobson-radical. Finally, in Section 4, we shall study rings all of whose proper factor rings are qc. Such rings will be called “ restricted qc ”.

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“…In particular, a non local indecomposable right q-ring is a left q-ring. [7]). The purpose of our paper is to extend this line of research by studying rings in which every right ideal is quasi-continuous (right %-rings).…”

mentioning

confidence: 99%

“…Rings for which every right ideal is quasi-injective (known as right q-rings) have been studied by several authors (c.f. [4], [5], [6], [7]). The purpose of our paper is to extend this line of research by studying rings in which every right ideal is quasi-continuous (right %-rings).…”

mentioning

confidence: 99%

Rings with quasi-continuous right ideals

1999

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Abstract. Rings in which each right ideal is quasi-continuous (right %-rings) are shown to be a direct sum of semisimple artinian square full ring and a right square free ring. Among other results it is also shown that (i) a nonlocal right continuous indecomposable right %-ring is either simple artinian or a ring of matrices of a certain type, and (ii) an indecomposable non-local right continuous ring is both a right and a left %-ring if and only if it is a right q-ring. In particular, a non local indecomposable right q-ring is a left q-ring. [7]). The purpose of our paper is to extend this line of research by studying rings in which every right ideal is quasi-continuous (right %-rings). In Section 2 we show that a %-ring is a direct sum of a semisimple artinian square full ring and a square free ring.In Section 3 we study right continuous right %-rings. We show that a non-local, right continuous, indecomposable right %-ring R is either simple artinian or a ring of matrices of a certain type (Theorem 3.8). We show that an indecomposable, nonlocal, right continuous ring R is both a right and a left %-ring if and only if R is a right q-ring (Theorem 3.13). In particular, under these hypotheses, R is a right q-ring if and only if it is a left q-ring.1. De®nitions and preliminaries. Throughout the paper R will be a ring with identity and all R-modules will be unital right R-modules, unless otherwise stated. For modules M, N, the notations N & H M and N & È M respectively serve to denote that N is an essential submodule of M and that N is a direct summand of M. By ZM, oM we denote the singular submodule and socle of M, respectively. JR is the Jacobson Radical of R.M (or EM) stands for the injective hull of M. rom R MY N stands for the set of R-homomorphisms from M to N. ind R M is the set of R-endomorphisms of M. If N & M, a closure of N in M is a maximal essential extension of N in M. N is said to be closed in M if N has no proper essential extension on M. Given right R-modules M and N, M is said to be N-injective i for any K & N, every P rom R KY M is the restriction of some P rom R NY M.For a module M we consider the following conditions.

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Rings in which every right ideal is quasi-injective

Cited by 39 publications

References 5 publications

A Generalization of Injectivity for Modules Over a Unitary Ring

A Generalization of Injectivity for Modules Over a Unitary Ring

On rings with quasi-injective cyclic modules

Rings with quasi-continuous right ideals

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