scholarly journals AGQP-Injective Modules

2008 ◽  
Vol 2008 ◽  
pp. 1-7
Author(s):  
Zhanmin Zhu ◽  
Xiaoxiang Zhang
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Injective Modules

Let be a ring and let be a right -module with = End(). is calledalmost general quasi-principally injective(orAGQP-injectivefor short) if, for any , there exist a positive integer and a left ideal of such that and . Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additional conditions are studied.

scholarly journals On (co)pure Baer injective modules

2021 ◽  
Vol 31 (2) ◽  
pp. 219-226
Author(s):  
M. F. Hamid ◽  
Keyword(s):  
Left Ideal ◽  
Injective Module ◽  
Injective Modules

For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as Q-copure submodule of a Q-copure Baer injective module. Certain types of rings are characterized using properties of Q-copure Baer injective modules. For example a ring R is Q-coregular if and only if every Q-copure Baer injective R-module is injective.

ON n-CENTRALIZING GENERALIZED DERIVATIONS IN SEMIPRIME RINGS WITH APPLICATIONS TO C*-ALGEBRAS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250111 ◽  
Author(s):  
BASUDEB DHARA ◽  
SHAKIR ALI
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
C Algebras ◽  
Semiprime Rings ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xn – xn f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

RELATIVE COHERENCE OF RINGS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250047
Author(s):  
LIXIN MAO ◽  
NANQING DING
Keyword(s):  
Left Ideal ◽  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Injective Modules ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring and τ a hereditary torsion theory for the category of all left R-modules. A right R-module M is called τ-flat if Tor 1(M, R/I) = 0 for any τ-finitely presented left ideal I. A left R-module N is said to be τ-f-injective in case Ext 1(R/I, N) = 0 for any τ-finitely presented left ideal I. R is called a left τ-coherent ring in case every τ-finitely presented left ideal is finitely presented. τ-coherent rings are characterized in terms of, among others, τ-flat and τ-f-injective modules. Some known results are extended.

A Note on GP-Injectivity

2009 ◽  
Vol 16 (04) ◽  
pp. 625-629
Author(s):  
Yasuyuki Hirano ◽  
Hong Kee Kim ◽  
Jin Yong Kim
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Nonzero Element ◽  
Positive Answer

New characteristic properties of left GPGV-rings are given. It is shown that if R is a left GPGV-ring, then for any nonzero element a in R, there is a positive integer n such that an≠ 0 and (RaR+ l(an))⊕ L=R for some left ideal L contained in Soc (RR). As a corollary of this result, we are able to give a positive answer to a question raised by Yue Chi Ming.

Generalized Derivations Having the Same Power Values with Left Multiplications

2013 ◽  
Vol 20 (03) ◽  
pp. 369-382 ◽  
Author(s):  
Xiaowei Xu ◽  
Jing Ma ◽  
Fengwen Niu
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Prime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Fixed Positive Integer

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal [Formula: see text] and a noncommutative Lie ideal L, respectively.

scholarly journals Generalizations of principally quasi-injective modules and quasiprincipally injective modules

2005 ◽  
Vol 2005 (12) ◽  
pp. 1853-1860 ◽  
Author(s):  
Zhu Zhanmin ◽  
Xia Zhangsheng ◽  
Tan Zhisong
Keyword(s):  
Left Ideal ◽  
Injective Modules

LetRbe a ring andMa rightR-module withS=End(MR). The moduleMis called almost principally quasi-injective (orAPQ-injective for short) if, for anym∈M, there exists anS-submoduleXmofMsuch thatlMrR(m)=Sm⊕Xm. The moduleMis called almost quasiprincipally injective (orAQP-injective for short) if, for anys∈S, there exists a left idealXsofSsuch thatlS(Ker(s))=Ss⊕Xs. In this paper, we give some characterizations and properties of the two classes of modules. Some results on principally quasi-injective modules and quasiprincipally injective modules are extended to these modules, respectively. Specially in the caseRR, we obtain some results onAP-injective rings as corollaries.

scholarly journals Weakly Projective and Weakly Injective Modules

1994 ◽  
Vol 46 (5) ◽  
pp. 971-981 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth ◽  
K. Oshiro ◽  
M. A. Saleh
Keyword(s):  
Positive Integer ◽  
Artinian Ring ◽  
Projective Cover ◽  
Finitely Generated ◽  
Matrix Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Qf Ring ◽  
Finitely Generated Modules

AbstractA module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:(1) R is a left perfect and every weakly projective right R-module is weakly injective.(2) R is a direct sum of matrix rings over local QF-rings.(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.(4) R is right artinian ring and every weakly injective right R-module is weakly projective.(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

scholarly journals Localizations of injective modules

1985 ◽  
Vol 28 (3) ◽  
pp. 289-299 ◽  
Author(s):  
K. R. Goodearl ◽  
D. A. Jordan
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Simple Module ◽  
Global Dimension ◽  
Injective Hull ◽  
Injective Dimension ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Noetherian Domain ◽  
Left And Right

The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

ON STRONGLY C2 MODULES AND D2 MODULES

2013 ◽  
Vol 12 (07) ◽  
pp. 1350029 ◽  
Author(s):  
WENXI LI ◽  
JIANLONG CHEN ◽  
FARID KOURKI
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Endomorphism Ring ◽  
Finitely Generated

Let R be a ring, MR be a right R-module, n be a positive integer and S = End (MR) be the endomorphism ring of MR. MR is called a strongly C2 module if [Formula: see text] is C2 for every positive integer m. MR is called an n-C2 module if the annihilator rM(K) ≠ 0 for any n-generated proper left ideal K of S. We prove that MR is strongly C2 if and only if M is n-C2 for every positive integer n, if and only if the annihilator rM(K) is not zero for every finitely generated proper left ideal K of S, and then we get some characterizations of right n-C2 rings and strongly right C2 rings. Also we obtain some dual statements of n-D2 module and strongly D2 module.

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