No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On quasi-injectivity and von Neumann regularity
Abstract
The purpose of this note is to consider certain connections between injectivity,p-injectivity and a generalisation of quasi-injectivity notedGQ-injectivity (cf. definition below). It is proved that ifA is a leftGQ-injective ring andZ the left singular ideal ofA, thenA/Z is von Neumann regular andZ is the Jacobson radical ofA (this extends the well-known result ofY. Utumi for left continuous rings [9]). If the sum of any twoGQ-injective leftA-modules isGQ-injective, thenA is a left Noetherian, left hereditary, leftV-ring. Semi-prime rings whose faithful left modules areGQ-injective must be semi-simple Artinian. IfA is commutative, the following are equivalent: (1)A is a finite direct sum of field; (2) EveryGQ-injectiveA-module is injective; (3) AnyA-module isGQ-injective if, and only if, it isp-injective; (4) AnyA-module is quasi-injective if, and only if, it isp-injective. Also, a commutative ringA is hereditary Noetherian if, and only if, the sum of any twop-injectiveA-modules is injective.
... . ■ Recall that a ring R is right GQ-injective [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then ...
... . ■ Recall that a ring R is right GQ-injective [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then ...
A ring is called right SNF-rings if every simple right R-module is N-flat . In this paper , we give some conditions which are sufficient or equivalent for a right SNF-ring to be n-regular (reduced) .It is shown that
1- If is a GW-ideal of R for every . then ,is reduced if and only if is right SNF-ring.
2- If is an reversible, then is regular if and only if is right GQ-injective and SSNF-ring .
... , which is a required contradiction . Therefore , [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then ...
... Therefore , [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then ...
A ring R is called right SNF-rings if every simple right R-module is N-flat . In this paper , we give some conditions which are sufficient or equivalent for a right SNF-ring to be n-regular (reduced) .It is shown that
1- If r(a) is a GW-ideal of R for every a R . then ,R is reduced if and only if R is right SNF-ring.
2- If R is an reversible, then R is regular if and only if R is right GQ-injective and
SSNF-ring .
... , which is a required contradiction . Therefore , [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then ...
... Therefore , [11] if , for any right ideal I isomorphic to a complement right ideal of R , every right R-homomorphism of I into R extends to an endomorphism of R . In [11] , shows that if R is right GQ-injective ring , then ...
A ring R is called right SNF-rings if every simple right R-module is N-flat . In
this paper , we give some conditions which are sufficient or equivalent for a right SNFring
to be n-regular (reduced) .It is shown that
1- If r(a) is a GW-ideal of R for every aR. then , R is reduced if and only if R is
right SNF-ring.
2- If R is an reversible, then R is regular if and only if R is right GQ-injective and
SSNF-ring .
... According to [16] ,a ring R is right GQ-injective if for any right ideal I isomorphic to a complement right ideal of R ,every right R-homomorphism of I into R extends to an ...
... In [16], ...
Let R be a ring. Let R M be a module with ( ) R S = End M .The module M is
called almost Wnil-injective (briefly right AWN-injective ) if , for any 0 ¹ aÎ N(R) ,there
exists n ³ 1 and an S-submodule a X of M such that an ¹ 0 and
an
n n
M R l (r (a )) = Ma Å X as left S-modules .If R R is almost Wnil-injective , then we call
R is right almost Wnil-injective ring . In this paper ,we give some characterization and
properties of almost Wnil-injective rings .In particular , Conditions under which right almost
Wnil-injective rings are n-regular rings and n-weakly regular rings are given .Also we study
rings whose simple singular right R-module are almost Wnil-injective , It is proved that if R is
a NCI ring ,MC2 , whose every simple singular R-module is almost Wnil-injective , Then R is
reduced .
... Let , then by Preposition 2.6, there exists a right ideal K of R such that then there exists such that so for all by Lemma (2.7), is flat right R-module. Y(R) is denoted to right singular ideal of R Lemma 2.9 [10] If then there exists such that Theorem 2. 10 Let R be a right quasi duo ring whose every simple singular right R-module is YJ-injective. Then R is right nonsingular ring. ...
... Let , then by Preposition 2.6, there exists a right ideal K of R such that then there exists such that so for all by Lemma (2.7), is flat right R-module. Y(R) is denoted to right singular ideal of R Lemma 2.9 [10] If then there exists such that Theorem 2. 10 Let R be a right quasi duo ring whose every simple singular right R-module is YJ-injective. Then R is right nonsingular ring. ...
In this paper , we give some characterizations and properties of Quasi duo rings whose every simple singular module is YJ-injective . and we study the relation between this rings and other rings , like NI-ring, non singular rings, generalized pi-regular ring, strongly regular and n-regular ring .
... Recall that a ring R is left GQ−injective [6] if, for any left ideal I isomorphic to a complement left ideal of R, every left R−homomorphism of I into R extends to an endomorphism of R R. In [6], Roger Yue Chi Ming shows that if R is left GQ−injective ring, then J(R) = Z l (R), R/J(R) is regular. ...
... Recall that a ring R is left GQ−injective [6] if, for any left ideal I isomorphic to a complement left ideal of R, every left R−homomorphism of I into R extends to an endomorphism of R R. In [6], Roger Yue Chi Ming shows that if R is left GQ−injective ring, then J(R) = Z l (R), R/J(R) is regular. ...
In this paper, we first study some characterizations of left MC2 rings. Next, by introducing left nil-injective modules, we discuss and generalize some well known results for a ring whose simple singular left modules are Y J-injective. Finally, as a byproduct of these results we are able to show that if R is a left MC2 left Goldie ring whose every simple singular left R-module is nil-injective and GJcp-injective, then R is a finite product of simple left Goldie rings.
... 3) Following [2] a ring R is called reduced if R has no nonzero nilpotent elements . 4) R is called uniform if every nonzero ideal of R is essential, see [3]. ...
... Then by Lemma 7 [3] , there exists ...
In this work, we study rings whose every principal ideal is a right pure. We give some properties of right PIP – rings and the connection between such rings and division rings.
... Recall that A is left continuous (in the sense of Y. Utumi) if every left ideal of A which is isomorphic to a complement left ideal is a direct summand of A A. In [32], left continuous rings are generalized as follows: A is a left GQ-injective ring if, for any left ideal C of A which is isomorphic to a complement left ideal of A, every left A-homomorphism of C into A extends to an endomorphism of A A. ...
... Suppose that Z = (0). By[32, Lemma 7], there exists z ∈ Z, z = 0 such that z 2 = 0. Let M be a maximal left ideal of A containing l(z). Then M is an essential left ideal of A and M is an ideal of A by hypothesis. ...
- A generalization of injectivity is studied and several properties are developed. Von Neumann regular rings are char- acterized. Sufficient conditions are given for a ring to admita strongly regular classical left quotient ring. A nice characteriza- tion of strongly regular rings is given. Special direct summands of left self-injective regular and left continuous regular rings are considered.
... The ring of integers module 6 , 6 Z is a non-singular ring . Recall the following result of Ming in [2]. ...
... The ring of integers module 6 , 6 Z is a non-singular ring . Recall the following result of Ming in [2]. ...
In this paper several new properties of singular ideals and non- singular rings are obtained, a connection between a singular ideal and the Jacobson radical is considered , and a sufficient condition for a non- singular ring to be reduced is given.
... It suffices to show that Y(R) = 0 by Lemma 2.13. Suppose that Y(R) 0, there exists a nonzero element a Y(R) such that a 2 = 0 [10]. Then, there exists a maximal right ideal M of R containing Y(R)+ r(a), whence M is an essential. ...
In this paper we investigate von Neumann regularity of rings whose simple singular right R-modules are flat. It is proved that a ring R is strongly regular if and only if R is a semiprime right quasi-duo ring whose simple singular right R-modules are flat. Moreover, it is shown that if R is a P. I.-ring whose simple singular right R-modules are flat, then R is a strongly p-regular ring. Finally, it is shown that if R is a right continuous ring whose simple singular right R-modules are flat, then R is a von Neumann regular ring.
... If Y(R) 0, by Lemma (7) of [6], there exists 0 y Y(R) with y 2 =0. Let L be a maximal right ideal of R, set L = y R+ r(y) , we claim that L is essential right ideal o f R . ...
The main purpose of this paper is to study ERT and MERT rings, in order to study the connection between such rings and II-regular rings.
... , by Lemma (7) of [6] , there exists ) ( 0 R Y y ∈ ≠ with y 2 = 0 . Let L be a maximal right ideal of R, containing r(y) .We claim that L is an essential right ideal of R. Suppose this is not true, then there exists a non-zero ideal T of R such that L I T = (0) . ...
MEP) ﺍﻟ ﻤﺜـﺎﻟﻲ ﺠﺯﺀ ﻜل ﻓﻴﻬﺎ ﺍﻟﺘﻲ ﺤﻠﻘﺎﺕ ﺍﻴﻤﻥ ﺃﻋﻅﻤﻲ ﺃﺴﺎﺴﻲ ﻨﻘﻲ ﻫﻭ ﺃﻴﺴﺭ (ﻭﺇﻋﻁﺎﺀ ﻟﻬـﺎ ﺍﻷﺴﺎﺴﻴﺔ ﺍﻟﺨﻭﺍﺹ. ﺍﻟـﺸﺭﻭﻁ ﺇﻋﻁـﺎﺀ ﻜـﺫﻟﻙ ﻟﻠﺤﻠﻘﺔ ﻭﺍﻟﻜﺎﻓﻴﺔ ﺍﻟﻀﺭﻭﺭﻴﺔ MEP ﺒﻀﻌﻑ ﻭﻤﻨﺘﻅﻤﺔ ﺒﻘﻭﺓ ﻤﻨﺘﻅﻤﺔ ﺤﻠﻘﺔ ﺘﻜﻭﻥ ﻟﻜﻲ. ABSTRACT This paper introduces the notion of a right MEP-ring (a ring in which every maximal essential right ideal is left pure) with some of their basic properties; we also give necessary and sufficient conditions for MEP-rings to be strongly regular rings and weakly regular rings.
... Following [11], a ring R is a right (left) weakly regular if I 2 =I for each right (left) ideal I of R . Equivalently, if a∈aRaR (a∈RaRa) for every a in R . ...
The purpose of this paper is to study the class of the rings for which every maximal right ideal is left GP-ideal. Such rings are called MGP-rings and give some of their basic properties as well as the relation between
MGP-rings, strongly regular ring, weakly regular ring and kasch ring.
... If Y(R) ≠0, by Lemma (7) of [6], there exists 0 ≠ y ∈ Y(R) with y 2 =0. Let L be a maximal right ideal of R, set L = y R+ r(y) , we claim that L is essential right ideal of R. Suppose this is not true, then there exists a non-zero ideal T of R such that L ∩ T =(0). ...
... Applying C [10], Lemma 73 and Lemma 2, we get ...
... Recall that R is a left GQ−injective ring [9] (2) Let R be a left GQ−injective ring whose simple singular left R−module is W nil−injective. Then R is a Von Neumann regular ring. ...
A ring R is called left nil-injective if every R-homomorphism from a principal left ideal which is generated by a nilpotent element to R is a right multiplication by an element of R. In this paper, we first introduce and characterize a left nil-injective ring, which is a proper generalization of left p-injective rings. Next, various properties of left nil-injective rings are developed, many of them extend known results.
... We claim that Y=0. Suppose that Y≠ 0 then by [5], there exists 0 ≠ y∈Y such that y 2 = 0. ...
A ring R is called a right SF-ring if all its simple right R-modules are flat. It is well known that Von Neumann regular rings are right and left SF-rings. In this paper we study conditions under which SF-rings are strongly regular. Finally, some new characterstic properties of right SF-rings are given.
... Left GQ-injective rings generalize left continuous rings in the sense of Utumi [9]. In [5] Ming proved that if R is left GQ-injective ring then J(R) = Z l (R) and R/J(R) is von Neumann regular. Recall that a left R-module M is left principally injective (briefly left P-injective) if, for any principal left ideal Ra of R, every left R-homomorphism from Ra into M extends to one from R into M . ...
The purpose of this note is to improve several known results on GQ-injective rings. We investigate in this paper the von Neumann regularity of left GQ-injective rings. We give an answer to a question of Yue Chi Ming in the positive. Actually it is proved that if R is a left GQ-injective ring whose simple singular left R-modules are GP-injective then R is a von Neumann regular ring.
... Recall that R is a left GQ−injective ring [9] ...
In this paper, the definition of right nil n-injective ring is given, it is a generalization of nil injective rings. The aim of this paper is to investigate properties of nil n-injective rings. Various results are developed, many extending known results.
As a generalization of regular rings, that is a ring is called n-regular if for all . In this paper, we first give various properties of n-regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI rings, and other types of rings.
ABSTRACT As a generalization of right weakly regular rings, we introduce the notion of right n-weakly regular rings, i.e. for all a N(R), aaRaR. In this paper, first give various properties of right n-weakly regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI, MC2 and SNF rings. Introduction:
ABSTRACT As a generalization of regular rings, that is a ring is called n-regular if aRa a ∈ for all) (R N a ∈. In this paper, we first give various properties of n-regular rings. Also, we study the relation between such rings and reduced rings by adding some types of rings, such as NCI rings, and other types of rings.
It is known that a ring R is left Noetherian if and only if every left R-module has an injective (pre)cover. We show that (1) if R is a right n-coherent ring, then every right R-module has an (n, d)-injective (pre)cover; (2) if R is a ring such that every (n, 0)-injective right R-module is n-pure extending, and if every right R-module has an (n, 0)-injective cover, then R is right n-coherent. As applications of these results, we give some characterizations of (n, d)-rings, von Neumann regular rings and semisimple rings.
We study weakly regular rings and some generalizations of V-rings via GW-ideals. We show that: (1) If R is a left weakly regular ring whose maximal left (right) ideals are GW-ideals, then R is strongly regular; (2) If R is a right weakly regular ring whose maximal essential left ideals are GW-ideals, then R is ELT regular; (3) If R is a ring in which l(a) is a GW-ideal for all a∈R, then R is left weakly regular if and only if R is right weakly regular; (4) A ring R is strongly regular if and only if R is a ZI left GP-V ' -ring whose maximal essential left (right) ideals are GW-ideals; (5) If R is a left (right) GP-V-ring such that l(a) is a GW-ideal for all a∈R, then R is weakly regular.
Let M1 and M2 be two R-modules. Then M2 is called M1-c-injective if every homomorphism α from K to M2, where K is a closed submodule of M1, can be extended to a homomorphism β from M1 to M2. An R-module M is called self-c-injective if M is M-c-injective. For a projective module M, it has been proved that the factor module of an M-c-injective module is M-c-injective if and only if every closed submodule of M is projective. A characterization of self-c-injective modules in terms endomorphism ring of an R-module satisfying the CM-property is given.
The following results are proved for a ring A: (1) If A is a fully right idem- potent ring having a classical left quotient ring Q which is right quasi-duo, then Q is a strongly regular ring; (2) A has a classical left quotient ring Q which is a finite di- rect sum of division rings iff A is a left TC-ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let A have the following properties: (a) each maximal left ideal of A is either a two-sided ideal of A or an injective left A-module; (b) for every maximal left ideal M of A which is a two-sided ideal, A/MA is flat. Then, A is either strongly regular or left self-injective regular with non-zero socle; (4) A is strongly regular iff A is a semi-prime left or right quasi-duo ring such that for every essential left ideal L of A which is a two-sided ideal, A/LA is flat; (5) A prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a YJ-injective ring with maximum condition on annihilators.
A generalization of semi-prime rings, calledSRN rings, is introduced. Reduced ideals, annihilators and various properties of semi-prime and self-injective rings are considered.
Brainerd and Lambek (2, Corollary 4) have proved recently that any complete Boolean ring is self-injective. It is easy to see that every complete Boolean ring is a continuous regular ring, that is, a regular ring of which the lattice of principal left ideals is continuous. This suggests that in a continuous regular ring it might be possible to prove the injectivity. However, a simple example (Example 3) shows that the conjecture is not true in general. Our main theorem is the following. Every continuous regular ring with no ideals of index 1 is (both left and right) self-injective (Theorem 3).
It is known to Wolfson (13, Theorem 5.1) and Zelinsky (15) that the ring S of all linear transformations of a vector space of dimension ≥ 2 over a division ring is generated by idempotents and also by non-singular elements.
Throughout, A denotes an associative ring with identity and “module” means “left, unitary A -module”. In ( 3 ), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A . It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see ( 2 , p. 130)). The second proposition here is a partial generalisation of that result.
Summary In this sequel to [14]and [15],a generalization of quasi-injective modules, noted FK-injective, is introduced to study von Neumann regular and continuous rings. This will lead to new characteristic properties of continuous regular rings. Conditions for certain non-singular modules to be completely reducible and injective are given. A few decompositions of FK-injective modules are considered.
This note contains certain conditions for unit-regularity and for selfinjective regular rings to have bounded index (such rings have many interesting properties [9]). Sufficient conditions are also given for rings to be Noetherian, Artinian and quasi-Frobeniusian.
This note is a natural sequel to [8] and [9]. Further characteristic properties of arbitrary von Neumann regular rings and strongly regular rings are given in terms of annihilators and simple modules. A prime ring with certain annihilator conditions is shown to be primitive (this is related to the following problem ofKaplansky: Are prime regular rings primitive?). Necessary and sufficient conditions for leftq-rings to be regular are also considered: For example, a leftq-ring is regular iff every simple rightA-module is flat. A sufficient condition is given for a leftqc-ring to be a uniserial, strongly left and strongly rightqc, left and rightq-ring. One of the main results ofJain, Mohamed andSingh onq-rings [5, Theorem 2.13] is generalised. Finally, it is shown that a prime left continuous ring either has zero socle or is primitive, left self-injective regular.
Injective modules.- Essential extensions and the injective hull.- Quasi-Injective modules.- Radical and semiprimitivity in rings.- The endomorphism ring of a quasi-injective module.- Noetherian, artinian, and semisimple modules and rings.- Rational extensions and lattices of closed submodules.- Maximal quotient rings.- Semiprime rings with maximum condition.- Nil and singular ideals under maximum conditions.- Structure of noetherian prime rings.- Maximal quotient rings.- Quotient rings and direct products of full linear rings.- Johnson rings.- Open problems.
Proposition 6] and [10, Proposition 3] to Proposition 1, we get Corollary 1.1. The following conditions are equivalent for a left G Q-injective ring: (1) A is regular
- Since
Since A is semi-simple iffAA/Jis flat, by applying [7, Proposition 6] and [10, Proposition 3] to Proposition 1, we get Corollary 1.1. The following conditions are equivalent for a left G Q-injective ring: (1) A is regular. (2) A is a right p-V-ring.
Every left A-module is CS. (5) A is a semi-prime ring whose faithful left modules are GQ-injective
- R Yue
R. YUE CHIMING (4) Every left A-module is CS. (5) A is a semi-prime ring whose faithful left modules are GQ-injective.