Comparing First Order Theories of Modules over Group Rings II: Decidability

We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T(RG) of all R‐torsionfree RG‐modules and the theory T₀(RG) of RG‐lattices (i. e. finitely generated R‐torsionfree RG‐modules), and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG‐lattices are of finite, or wild representation type.     

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 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.     

Modular Adjacency Algebras of Hamming Schemes

To each association scheme G and to each field R, there is associated naturally an associative algebra, the so-called adjacency algebra RG of G over R. It is well-known that RG is semisimple if R has characteristic 0. However, little is known if R has positive characteristic. In the present paper, we focus on this case. We describe the algebra RG if G is a Hamming scheme (and R a field of positive characteristic). In particular, we show that, in this case, RG is a factor algebra of a polynomial ring by a monomial ideal.     

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

On Schur classes for modules over group rings

We consider the problem of coupling between a quotient module A/C _A (G) and a submodule A(ωRG), where G is a group, R is a ring, and A is an RG-module; C _A (G) can be considered as an analog of the center of the group, and the submodule A(ωRG) can be considered as an analog of the derived subgroup of the group. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1261–1268, September, 2007.     

Generalized twisted group rings of finite groups

Let R be a discrete valuation ring and G a finite group. Let α: G × G → R\{0} be a generalized 2-cocycle, i.e. not necessarily taking its values in the units of R, and consider the generalized twisted group ring R*_α G. In this paper we derive a criterion for R*_α G. to be separable over its center. This leads us to study the center and the prime ideals of R*_α G. Furthermore, we give some results on indecomposable modules over R*_α G.     

Rugged modules: The opposite of flatness

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M⁺ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.     

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.     

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)?{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.     

Duality for a Cohen–Macaulay Local Ring

Let (R, 𝔪) be a Cohen–Macaulay local ring. If R has a canonical module, then there are some interesting results about duality for this situation. In this article, we show that one can indeed obtain similar results in the case R does not have a canonical module. Also, we give some characterizations of complete big Cohen–Macaulay modules of finite injective dimension, and by using them, some characterizations of Gorenstein modules over the 𝔪-adic completion of R are obtained.     

Torsion Endo-Trivial Modules

We prove that the group T(G) of endo-trivial modules for a noncyclic finite p-group G is detected on restriction to the family of subgroups which are either elementary Abelian of rank 2 or (almost) extraspecial. This result is closely related to the problem of finding the torsion subgroup of T(G). We give the complete structure of T(G) when G is dihedral, semi-dihedral, or quaternion.     

Finitistic n-Self-Cotilting Modules

We study a class of modules which can be characterized using a duality theorem, called finitistic n-self-cotilting. Such a module Q can be characterized using dual conditions of some generalizations for star modules: every module M which has a right resolution with n terms isomorphic to finite powers of Q (i.e., M is n-finitely Q-copresented) has a right resolution with (n + 1) terms, and the functor Hom _R (?, Q) preserves the exactness of all monomorphisms with their ranges finite powers of Q and cokernels n-finitely Q-copresented modules. In the general case, these modules are independent toward other kinds of modules which are characterized using some dualities (w-Π _f -quasi injective modules, costar modules, f-cotilting modules). Closure properties for the classes involved in the duality are studied. At the end of the article, connections with the cotilting theory are exhibited, in the case of finitely dimensional algebras over fields.     

Flat and Stably Flat Modules

We study Kropholler's generalisation of Lazard's criterion for a module to be flat. The context is complete cohomology and modules of type (FP)_∞. Let G be a group in the class ₁ of groups which act on a finite dimensional contractible cell complex with finite stabilisers and let R be a strongly G-graded algebra. We provide a characterisation of the stably flat R-modules under the assumption that R₁ is coherent of finite global dimension.     

Notes on Divisible and Torsionfree Modules

In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied.     

AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

We investigate the properties of categories of G _C -flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G _C -flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R-modules of finite G _C -flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G _C -flat R-modules yield only the modules in the original subcategories.     

Some Decidability Results for ℤ[G]-Modules when G is Cyclic of Squarefree Order

We extend the analysis of the decision problem for modules over a group ring ?[G] to the case when G is a cyclic group of squarefree order. We show that separated ?[G]-modules have a decidable theory, and we discuss the model theoretic role of these modules within the class of all ?[G]-modules. The paper includes a short analysis of the decision problem for the theories of (finitely generated) modules over ?[ζ_m], where m is a positive integer and ζ_m is a primitive mth root of 1. Mathematics Subject Classification: 03C60, 03B25.     

Symmetric Elements Under Oriented Involutions in Group Rings

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ? of the group G, an oriented invoultion ^σ? of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.