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1.
A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Ran into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].  相似文献   

2.
Gorenstein flatness and injectivity over Gorenstein rings   总被引:1,自引:0,他引:1  
Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.  相似文献   

3.
In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.  相似文献   

4.
Let R be a ring. R is called right AP-injective if, for any a ∈ R, there exists a left ideal of R such that lr(a) = Ra (?) Xa. We extend this notion to modules. A right .R-module M with 5 = End(MR) is called quasi AP-injective if, for any s ∈ S, there exists a left ideal Xs of S such that ls(Ker(s)) = Ss (?) Xs. In this paper, we give some characterizations and properties of quasi AP-injective modules which generalize results of Page and Zhou.  相似文献   

5.
Global dimension and left derived functors of Hom   总被引:1,自引:0,他引:1  
It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R)≤n (n≥2) if and only if the gl right Proj-dim MR≤n - 2 if and only if Extn-1(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R)≤n (n≥1) if and only if every (n-1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R)≤2 are characterized.  相似文献   

6.
Let R be a ring with unit element, and A a left R-module. A is called FP-injective if ExtR′(P, A)=0 for every finitely presented R-module P. Let ER(A) denote the injective hull of A. In the paper we prove by means of inverse limit functor that any module A Over coherent ring R has the minimum FP-injective submodule eR(A) in ER(A) which contains A and can be defined as the FP-injective hull of A.  相似文献   

7.
Let U be a flat right R-module and N an infinite cardinal number.A left R-module M is said to be (N,U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (N,U)-finitely presented in σ[M].It is proved under some additional conditions that a left R-module M is (N,U)-coherent if and only if Л^Ni∈I U is M-flat as a right R-module if and only if the (N,U)-coherent dimension of M is equal to zero.We also give some characterizations of left (N,U)-coherent dimension of rings and show that the left N-coherent dimension of a ring R is the supremum of (N,U)-coherent dimensions of R for all flat right R-modules U.  相似文献   

8.
Let R be a left and right Noetherian ring and n,k be any non-negative integers.R is said to satisfy the Auslander-type condition G n (k) if the right flat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 i n 1.In this paper,we prove that R is Gn (k) if and only if so is a lower triangular matrix ring of any degree t over R.  相似文献   

9.
In basic homological algebra,the flat and injective dimensions of modules play an important and fundamental role.In this paper,the closely related I F P-flat and I F P-injective dimensions are introduced and studied.We show that I F P-fd(M) = I F P-id(M +) and I F P-fd(M +)=I F P-id(M) for any R-module M over any ring R.Let IIn(resp.,IFn) be the class of all left(resp.,right) R-modules of IFP-injective(resp.,I F P-flat) dimension at most n.We prove that every right R-module has an IF npreenvelope,(IF n,IF n⊥) is a perfect cotorsion theory over any ring R,and for any ring R with I F P-id(RR) ≤ n,(II n,II n⊥) is a perfect cotorsion theory.This generalizes and improves the earlier work(J.Algebra 242(2001),447-459).Finally,some applications are given.  相似文献   

10.
关于拟AP内射模的注记   总被引:2,自引:0,他引:2  
赵玉娥  杜先能 《东北数学》2006,22(4):433-440
Let R be a ring.A right R-module M with S=End(MR)is called aquasi AP-injective module,if,for any s ∈ S,there exists a left ideal X_s of S such thatl_s(ker s)=SsX_s.Let M be a quasi AP-injective module which is a self-generator.We show that for such a module,if S is semiprime,then every maximal kernel of S isa direct summand of M.Furthermore,if ker(a_1)ker(a_2a_1)ker(a_3a_2a_1)satisfy the ascending conditions for any sequence a_1,a_2,a_3,...∈ S,then S is rightperfect.In this paper,we give a series of results which extend and generalize resultson AP-injective rings.  相似文献   

11.
R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:FM such that given any torsion-free moduleG and λ∈Hom R (G, M) there exists μ∈Hom R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.  相似文献   

12.
John Clark  Rachid Tribak 《代数通讯》2013,41(11):4390-4402
An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R R is F-almost injective.  相似文献   

13.
Linear topology defined on an arbitrary right module over a right Noetherian serial ring R enables one to describe the reduced, pure injective R-modules as modules that are complete in this topology. With the use of the completion of modules, the pure injective envelope of any right R-module is constructed. Bibliography: 8 titles.  相似文献   

14.
It is proved that any right module over a serial, right Noetherian ring R contains a basic submodule. The structure of submodules of an indecomposable pure injective R-module is investigated. Bibliography: 11 titles.  相似文献   

15.
Extending the notion of von Neumann regular elements in a ring R, a homomorphism f:AM between R-modules is said to be regular if there exists some g:M → A such that fgf = f. In this paper we report about the use of this term in module theory.   相似文献   

16.
Torsion-free Abelian groups G and H are called quasi-equal (GH) if λGHG for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if R M is an arbitrary unitary left module and M + is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M +) under the canonical ring homomorphism RE(M +). Then it holds that if E(M+) N is a pure submodule in E(M+) M +, then R N is a pure submodule in R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.  相似文献   

17.
We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.  相似文献   

18.
In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG nG n/Hom R (M, G) is an injective precover if Ext R 1 (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depthR R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depthR R > 0.  相似文献   

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