When Some Complement of a Submodule Is a Summand

A module M is said to satisfy the C ₁₁ condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C ₁ condition implies the C ₁₁ condition and that the class of C ₁₁-modules is closed under direct sums but not under direct summands. We show that if M = M ₁ ⊕ M ₂, where M has C ₁₁ and M ₁ is a fully invariant submodule of M, then both M ₁ and M ₂ are C ₁₁-modules. Moreover, the C ₁₁ condition is shown to be closed under formation of the ring of column finite matrices of size Γ, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C ₁₁ are essential extensions of C ₁₁-modules constructed from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C ₁₁-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.     

Two Open Questions On Goldie Extending Modules

Five open questions on Goldie extending modules were posed by Akalan et al. [1 Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The first one and the second one are considered in this article. It is shown that a 𝒢-extending module M with (C ₃) is 𝒢⁺-extending. Moreover, let M = M ₁ ⊕ M ₂ be a direct sum of 𝒢-extending modules, where M satisfies (C ₃) and M ₁ is M ₂-ojective (or M ₂ is M ₁-ojective), then M is 𝒢-extending. Other sufficient conditions for a direct sum of 𝒢-extending modules to be 𝒢-extending are obtained.     

Corrigendum To: Goldie Extending Modules

This corrigendum is written to correct the proof of Theorem 5.3 of Akalan et al. [1 Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

On Goldie Extending Modules with Condition (C2)

A module M is said to be continuous if it is extending with the condition (C₂) (cf. [6 Mohamed, S. H., Müller, B. J. (1999). Continuous and Discrete Modules. London Math. Soc. LNS, Vol. 147. Cambridge: Cambridge Univ. Press. [Google Scholar]], [7 Oshiro, K. (1983). Continuous modules and quasi-continuous modules. Osaka J. Math. 20:681–694.[Web of Science ®] , [Google Scholar]]). In this article, we consider a 𝒢-extending module with (C₂) which is a generalization of a continuous module. First, we show that any 𝒢-extending module with (C₂) satisfies the exchange property. We also prove that, if M₁ and M₂ are 𝒢-extending modules with (C₂), then M₁ ⊕ M₂ is 𝒢-extending with (C₂) if and only if M_i is M_j-ejective (i ≠ j).     

Modules Whose t-Closed Submodules Have a Summand as a Complement

We define and investigate T ₁₁-type modules as a generalization of t-extending modules, and modules satisfying C ₁₁ condition. A module M is said to be T ₁₁-type if every t-closed submodule of M has a complement which is a direct summand. Direct sums of T ₁₁-type modules inherit the property. Some equivalent conditions for a module M to be T ₁₁-type are given. We characterize a module M for which every direct summand satisfies T ₁₁ condition. If R _R is T ₁₁-type, then R/Z ₂(R _R ) is a C ₂ ring if and only if it is a von Neumann regular ring. Applying this result, we characterize a right t-extending (resp., finitely Σ-t-extending, or Σ-t-extending) ring R for which R/Z ₂(R _R ) is von Neumann regular.     

When some complement of a z-closed submodule is a summand

In this article we study modules with the condition that every z-closed submodule has a complement which is a direct summand. This new class of modules properly contains the class of extending modules. It is well known that the class of extending modules is closed under direct summands, but not under direct sums. In contrast to extending (or CS) modules, it is shown that the class of modules with former property is closed under direct sums. However we provide number of algebraic topological examples which show that this new class of modules is not closed under direct summands. To this end we obtain several results on the inheritance of the latter closure property.     

t-Extending Modules and t-Baer Modules

We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z ₂(R _R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective.     

Relative Projective Modules and Relative Injective Modules

Let R be a ring, and n and d fixed non-negative integers. An R-module M is called (n, d)-injective if Ext ^d+1 _R (P, M) = 0 for any n-presented R-module P. M is said to be (n, d)-projective if Ext¹ _R (M, N) = 0 for any (n, d)-injective R-module N. We use these concepts to characterize n-coherent rings and (n, d)-rings. Some known results are extended.     

Gorenstein Injective and Projective Complexes with Respect to a Semidualizing Module

Let R be a commutative (possibly non-Noetherian) ring (in order to make things less technical) and C a semidualizing R-module. In this article, we introduce and investigate the notion of G _C -injective (G _C -projective) complexes. This extends Enochs and García Rozas's notion of Gorenstein injective (Gorenstein projective) complexes. We then show that a complex X is G _C -injective (G _C -projective) if and only if X ^m is a G _C -injective (G _C -projective) module for each m ∈ ?.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Weak Equality and Exact Sequences in H v -modules

The largest class of multivalued systems satisfying the module-like axioms is the H_v-module. H_v-modules first were introduced by Vougiouklis. In this paper we define weak equality between two subsets of an H_v-module and introduced the notion of exact sequences of H_v-modules. Also some results on the weak equality and exact sequences are given.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

Generalized <Emphasis Type="Italic">SV</Emphasis>-modules

Given an arbitrary quasiprojective right R-module P, we prove that every module in the category σ(P) is weakly regular if and only if every module in σ(M/I(M)) is lifting, where M is a generating object in σ(P). In particular, we describe the rings over which every right module is weakly regular.     

When is the SIP (SSP) property inherited by free modules

Summary A right R-module M has right SIP (SSP) if the intersection (sum) of two direct summands of M is also a direct summand. It is shown that the right SIP (SSP) is not a Morita invariant property and that a nonsingular C₁₁⁺-module does not necessarily have SIP. In contrast, it is shown that the direct sum of two copies of a right Ore domain has SIP as a right module over itself.     

Extensions of <Emphasis Type="Italic">GM</Emphasis>-rings

It is shown that a ring R is a GM-ring if and only if there exists a complete orthogonal set { e ₁,...,e _n } of idempotents such that all e _i Re _i are GM-rings. We also investigate GM-rings for Morita contexts, module extensions and power series rings.This work was supported by the Natural Science Foundation of Zhejiang Province.     

Induced Projective Modules Over Group von Neumann Algebras

Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet. 71 (1996), 453–462; Arch. Math. 76 (2001), 241–249] and Schafer [K-theory 19 (2000), 211–217], while reducing to Swan’s theorem on induced representations [Ann. Math. 71 (1960), 552–578], in the case where the group G is finite. (Received: January 2005) Research supported by University of Athens grant Pythagoras 70/3/7298.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Weakly G K -Perfect and Integral Closure of Ideals

Let R be a commutative Noetherian ring, K a nonzero finitely generated suitable R-module, and I an ideal of R. It is shown that if (R, ) is local, then  is G _K -perfect if and only if K is a canonical module for R. Furthermore, if I is integrally closed and G _K  ? dim _R I < ∞, then K _ is a canonical R _-module for every  ? Ass _R R/I whenever K satisfies Serre's condition (S ₁) or grade _K I > 0. Finally, it is shown that if CM ? dim _R I < ∞, then R _ is Cohen–Macaulay for every  ? Ass _R R/I.