On Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G ₁ → G ₀ → G ⁰ → G ¹ → … such that the complex Hom _R (G, H) is exact for each projective left R-module H, the module Im(G ₀ → G ⁰) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.     

On Galois Comodules

Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P _A is finitely generated and projective and the evaluation map μ𝒞:Hom ^𝒞 (P, 𝒞) ?  _S P → 𝒞 is an isomorphism (of corings) where S = End ^𝒞 (P). It has been observed that for such comodules the functors ? ?  _A 𝒞 and Hom _A (P, ?) ?  _S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define Galois comodules without requiring P _A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

Monomorphism Operator and Perpendicular Operator

For a quiver Q, a k-algebra A, and an additive full subcategory 𝒳 of A-mod, the monomorphism category Mon(Q, 𝒳) is introduced. The main result says that if T is an A-module such that there is an exact sequence 0 → T _m  → … → T ₀ → D(A _A ) → 0 with each T _i  ∈ add(T), then Mon(Q, ^⊥ T) =^⊥(kQ ? _k T); and if T is cotilting, then kQ ? _k T is a unique cotilting Λ-module, up to multiplicities of indecomposable direct summands, such that Mon(Q, ^⊥ T) =^⊥(kQ ? _k T).

As applications, the category of the Gorenstein-projective (kQ ? _k A)-modules is characterized as Mon(Q, 𝒢𝒫(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, 𝒳) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.     

Butler Modules Over Prüfer Domains

If R is an integral domain, let  be the class of torsion free completely decomposable R-modules of finite rank. Denote by  the class of those torsion-free R-modules A such that A is a homomorphic image of some C ? , and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ? . Further, let Q  and Q 𝒫 be the respective closures of  and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q  = Q 𝒫, and  = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ?  has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then  = Q  = Q 𝒫 = 𝒫 if and only if R is almost maximal.     

Coherent Rings and Gorenstein FP-Injective Modules

In this article, Gorenstein FP-injective modules are introduced and investigated. A left R-module M is called Gorenstein FP-injective if there is an exact sequence … → E ₁ → E ₀ → E ⁰ → E ¹ → … of FP-injective left R-modules with M = ker(E ⁰ → E ¹) such that Hom _R (P, ?) leaves the sequence exact whenever P is a finitely presented left R-module with pd _R (P) < ∞. Some properties of Gorenstein FP-injective modules are obtained. Several well-known classes of rings are characterized in terms of Gorenstein FP-injective modules.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or _R ?, the category of left modules over a ring R.     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Generalizations of Groups in which Normality Is Transitive

A group G is called a Hall_𝒳-group if G possesses a nilpotent normal subgroup N such that G/N′ is an 𝒳-group. A group G is called an 𝒳_o-group if G/Φ(G) is an 𝒳-group. The aim of this article is to study finite solvable Hall_𝒳-groups and 𝒳_o-groups for the classes of groups 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯. Here 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯 denote, respectively, the classes of groups in which normality, permutability, and Sylow-permutability are transitive relations. Finite solvable 𝒯-groups, 𝒫𝒯-groups, and 𝒫𝒮𝒯-groups were globally characterized, respectively, in Gaschütz (1957 Gaschütz , W. ( 1957 ). Gruppen, in denen das normalteilersein transitiv ist . J. Reine Angew. Math. 198 : 87 – 92 .[Crossref] , [Google Scholar]), Zacher (1964 Zacher , G. ( 1964 ). I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali . Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 37 : 150 – 154 . [Google Scholar]), and Agrawal (1975 Agrawal , R. K. ( 1975 ). Finite groups whose subnormal subgroups permute with all Sylow subgroups . Proc. Amer. Math. Soc. 47 : 77 – 83 .[Crossref], [Web of Science ®] , [Google Scholar]). Here we arrive at similar characterizations for finite solvable Hall_𝒳-groups and 𝒳_o-groups where 𝒳 ∈ {𝒯, 𝒫𝒯, 𝒫𝒮𝒯}. A key result aiding in the characterization of these groups is their possession of a nilpotent residual which is a nilpotent Hall subgroup of odd order. The main result arrived at is Hall_𝒫𝒮𝒯 = 𝒯_o for finite solvable groups.     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

The Mal'cev Rank for Modules over Pseudo-Valuation Domains

Let R be a commutative ring and A be an R-module. The Mal'cev rank μ(A) of A is the sup of genN, where N ranges over the finitely generated submodules of A, and genN is the minimum number of generators of N. We prove that μ is both sub-additive and pre-additive as an invariant of Mod(R). Our main goal is to investigate μ for modules over pseudo-valuation domains. Specifically, we establish which pseudo-valuation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class 𝒞 of pseudo-valuation domains as a union 𝒞 = 𝒞₁ ∪ 𝒞₂ ∪ 𝒞₃ ∪ 𝒞₄ of suitably defined subclasses, and prove that the property holds if and only if R ∈ 𝒞₃ ∪ 𝒞₄. In that case we can describe the R-modules A where μ(A) < ∞. We also show that, for R ∈ 𝒞₄, there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.     

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Gorenstein coresolving categories

Let 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8 Enochs, E. E., Jenda, O. M. G. (1995). Gorenstein injective and projective modules. Math. Z. 220:611–633.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein FP-injective modules [20 Mao, L. X., Ding, N. Q. (2008). Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7:491–506.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein AC-injective modules [3 Bravo, D., Gillespie, J. (2016). Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44:2213–2233.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢?_𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢?_𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.     

Cotorsion Theories Cogenerated by a Torsionfree Class

Let R be a right perfect ring, and let (?, 𝒞) be a cotorsion theory in the category of right R-modules ? _R . In this article, it is shown that every right R-module has a superfluous ?-cover if and only if there exists a torsion theory (𝒜, ?) such that (?, 𝒞) is cogenerated by ?. It is also proved that if (𝒜, ?) is a cosplitting torsion theory, then (^⊥?, (^⊥?)^⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ?) is a centrally splitting torsion theory, then (^⊥?, (^⊥?)^⊥) is a hereditary and perfect cotorsion theory.     

Nonlinear Strong Commutativity Preserving Maps on Prime Rings

Let 𝒜 be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: 𝒜 → 𝒜 is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form Φ(A) = αA + f(A) for all A ∈ 𝒜, where α ∈ {1, ?1} and f is a map from 𝒜 into 𝒵(𝒜). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained.     

Lazard’s theorem for S-pure flat modules

Let R be an associative ring with identity. For a given class 𝒮 of finitely presented left (respectively right) R-modules containing R, we present a complete characterization of 𝒮-pure injective modules and 𝒮-pure flat modules. Consider that 𝒮 is a class of (R,R)-bimodules containing R with the following property: every element of 𝒮 is a finitely presented left and right R-module. We give a necessary and sufficient condition for 𝒮 to have Lazard’s theorem, and then we present our desired Lazard’s theorem.