Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢? _C  ? id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

On the Factorial Closure of Certain Monomial Modules

Abstract

Let R = K[y ₁,…,y _t ] be an affine domain over a field K and I be a nonzero proper ideal of R. In Sec. 1 of this note, we characterize when (K + I, R) is a Mori pair. In Sec. 2 of this note, we prove the following theorem: Let A ? B be domains such that C/Q is Mori for each subring C of B containing A and for any prime ideal Q of C. Then dim A ? 1 ≤ dim B ≤ dim A + 1 and if dim A > 1 or dim B > 1 then dim A = dim B.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

On Relative K 2-Group of Split Radical Ideals

Let I be a split radical ideal of a ring R. In this article, the exact sequence 1 → K ₂(R, I) → U _R (I) → V(R, I) → 1 is given by using the method of extension of groups, where U _R (I) is determined by generators and relations. The results of Maazen and Stienstra on the presentation for relative K ₂ group of split radical pairs are extended and amplified.     

Simple-Baer Rings and Minannihilator Modules

Let R be a ring. M is said to be a minannihilator left R-module if r _M l _R (I) = IM for any simple right ideal I of R. A right R-module N is called simple-flat if Nl _R (I) = l _N (I) for any simple right ideal I of R. R is said to be a left simple-Baer (resp., left simple-coherent) ring if the left annihilator of every simple right ideal is a direct summand of _R R (resp., finitely generated). We first obtain some properties of minannihilator and simple-flat modules. Then we characterize simple-coherent rings, simple-Baer rings, and universally mininjective rings using minannihilator and simple-flat modules.     

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 ∈ ?.     

Fiber Coefficients and Depth of Fiber Cones

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R, and K an ideal containing I. When r(I | K)<∞, we give a lower bound and an upper bound for f ₁(I). Under the above assumption on r(I | K) and depth G(I) ≥ d ? 1, we also provide a characterization, in terms of f ₁(I), of the condition depth F _K (I) ≥ d ? 1.     

A Note on Relative Flatness and Coherence

Let R be a ring and M a fixed right R-module. A new characterization of M-flatness is given by certain linear equations. For a left R-module F such that the canonical map M? _R F → Hom _R (M?, F) is injective, where M? = Hom _R (M, R), the M-flatness of F is characterized via certain matrix subgroups. An example is given to show that R need not be M-coherent even if every left R-module is M-flat. Moreover, some properties of M-coherent rings are discussed.     

On Isolated Submodules

Let R be a ring with identity and let M be a unital left R-module. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N ? K but L ? K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N ∩ IM = IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M.     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).     

On finitely generated module whose first nonzero fitting ideal is maximal

Let R be a Noetherian ring and M be a finitely generated R-module. Let I(M) be the first nonzero Fitting ideal of M. The main result of this paper asserts that when I(M) = Q is a regular maximal ideal of R, then M?R∕Q⊕P, for some projective R-module P of constant rank if and only if T(M)?QM. As a consequence, it is shown that if M is an Artinian R-module and I(M) = Q is a regular maximal ideal of R, then M?R∕Q.     

δ-M-Small and δ-Harada Modules

Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N? K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ _M -lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained.     

Simple-Direct-Projective Modules

In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a right R-module M is called simple-direct-projective if, whenever A and B are submodules of M with B simple and M/A ? B ?^⊕M, then A ?^⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ring R is artinian and serial with J²(R) = 0 if and only if every simple-direct-projective right R-module is quasi-projective if and only if every simple-direct-projective right R -module is a D3-module. It is also shown that a ring R is uniserial with J²(R) = 0 if and only if every simple-direct-projective right R-module is a C3-module if and only if every simple-direct-injective right R -module is a D3-module.     

Representation-Directed Incidence Coalgebras of Intervally Finite Posets and the Tame-Wild Dichotomy

Incidence coalgebras C = K ^□ I of intervally finite posets I that are representation-directed are characterized in the article, and the posets I with this property are described. In particular, it is shown that the coalgebra C = K ^□ I is representation-directed if and only if the Euler quadratic form q _C : ?^(I) → ? of C is weakly positive. Every such a coalgebra C is tame of discrete comodule type and gl. dimC ≤ 2. As a consequence, we get a characterization of the incidence coalgebras C = K ^□ I that are left pure semisimple in the sense that every left C-comodule is a direct sum of finite dimensional subcomodules. It is shown that every such coalgebra C = K ^□ I is representation-directed and gl. dimC ≤ 2. Finally, the tame-wild dichotomy theorem is proved, for the coalgebras K ^□ I that are right semiperfect.