Let R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S n (T + U) = SnT+SnUholds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited.

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called distributive if for every submodules S, T and U of M, the equality S boolean AND(T+U)=S boolean AND T+S boolean AND U holds true. In this paper, we give a necessary and sufficient condition for M to be distributive based on injective envelopes. The proof uses Matlis' results on injective modules.

Homology theory relative to classes of objects other than those of projective or injective objects in abelian categories has been widely studied in the last years, giving a special relevance to Gorenstein homological algebra. We prove the existence of Gorenstein flat precovers in any locally finitely presented Grothendieck category in which the class of flat objects is closed under extensions, the existence of Gorenstein injective preenvelopes in any locally noetherian Grothendieck category in which the class of all Gorenstein injective objects is closed under direct products, and the existence of special Gorenstein injective preenvelopes in locally noetherian Grothendieck categories with a generator lying in the left orthogonal class to that of Gorenstein injective objects.

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=SxT, 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.

We define Tate-Betti and Tate-Bass invariants for modules over a commutative noetherian local ring R. Then we show the periodicity of these invariants provided that R is a hypersurface. In case R is also Gorenstein, we show that a finitely generated R-module M and its Matlis dual have the same Tate-Betti and Tate-Bass numbers.

Given a double complex X there are spectral sequences with the E2 terms being either HI (HII(X)) or HII(HI(X)). But if HI(X) = HII(X) = 0 both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article we give a different way of constructing homology groups of X when HI(X) =HII(X) = 0. With this result we give a new and elementary proof of balance of Tate homology and cohomology. 1.Introduction We will mainly be concerned with left R-modules over some ring R. So unless otherwise specified, the term module will mean a left R-module. By a complex (C, d) of left R-modules we mean a graded module C = (Cn)n∈Z along with a morphism d = d : C → C of graded modules of degree −1 such that d ◦ d = 0. We also use the notation C = (C) but where d is of degree +1 and where we let Cn = C . Given a complex C we let Z(C) ⊂ C be Ker(d), let B(C) = Im(d) and let H(C) = Z(C)/B(C). If M and N are modules and C = (Ci) and D = (D ) are complexes, we form complexes denotedHom(M,D) andHom(C,N) whereHom(M,D) = Hom(M,D) and where Hom(C,N) = Hom(Ci, N). By a double complex of modules X we mean a bigraded module (X)(i,j)∈Z×Z along with morphisms d and d of bidegrees (1, 0) and (0, 1) respectively such that d◦d = 0, 2010 Mathematics Subject Classification.55U15,16E05,16E30, 18G15.

Formal power series come up in several areas such as formal language theory , algebraic and enumerative combinatorics, semigroup theory, number theory etc. This paper focuses on the set x R[[x]] consisting of formal power series with zero constant term. This subset forms a monoid with the composition operation on series. We classify the sets T of strictly positive integers for which the set of formal power series, R[[x^T]]={all formal power series consisting of terms whose power is from T}, forms a monoid with composition as the operation. We prove that in order for R[[x^T]] to be a monoid, T itself has to be a submonoid of N. Unfortunately, this condition is not enough to guarantee the desired result. But if a monoid is strongly closed, then we get the desired result. We also consider an analogous problem for power series in several variables.

We prove that, if F is the class of torsion free discrete modules over a profinite group G, that is, the class of discrete G-modules which are torsion free as abelian groups, then (F; F-perpendicular to) is a complete cotorsion pair. Moreover, we find a structure theorem for torsion free and cotorsion discrete G-modules and for finitely generated cotorsion discrete G-modules.

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.

Let R be a local commutative n-Gorenstein ring. The existence of the Gorenstein projective preenvelopes for finite R-modules is known (it was proved using duality arguments). In the present article, we compute an explicit Gorenstein projective preenvelope and a right Gorenstein projective resolution of a finite R-module. In light of this knowledge, we consider left derived functors , and Gexti(−, −). We prove a balance result for the Tate derived functor . Finally, we get an exact sequence connecting these derived functors.

When computing derived functors of a functor of two variables there is often balance. This means that these functors can be computed using a resolution of either variable. Now that flat covers are known to exist in many categories, derived functors can be computed using flat resolutions. So it is natural to ask if we have balance in this situation. When the ring is left perfect we get the usual balance. This raises the question of whether these are the only rings where we get balance. In this article we show that if the ring R is commutative and noetherian, then R must be perfect in order to get the desired balance.

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves.In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.

A ring is left Gorenstein regular if the classes of left modules with finite projective dimension and finite injective dimension coincide and the injective and projective finitistic left dimensions are finite. Let A and B be rings and U a (B,A)-bimodule such that UB has finite projective dimension and UA has finite flat dimension. In this paper we characterize when the ring T=(A0UB) is left Gorenstein regular and, over such rings, when a left T-module is Gorenstein projective or Gorenstein injective. As applications of these results, we characterize when T is left CM-free and give a necessary condition for existence of an infinite cardinal λ such that each Gorenstein projective module is a direct sum of λ<-generated modules.

We will show the interlacing between complete cotorsion pairs, model structures and homotopy categories. This will give a method of constructing adjoint functors between homotopy categories as well as a method for constructing abelian model structures in the category of unbounded complexes of certain abelian categories. We illustrate our methods by recovering some recent results of [23, 24, 25] as particular instances. And we also find new abelian model structures both in C(R) and in C(Sigma co(X)) attained to classes which are non necessarily closed under direct limits.

In this article, we consider the class of flat G-modules in the category of discrete modules over a profinite group G. We will appeal to a recent result of Enochs to prove that we have flat covers in this situation.

The idea of "vertex at the infinity" naturally appears when studying indecomposable injective representations of tree quivers. In this paper we formalize this behavior and find the structure of all the indecomposable injective representations of a tree quiver of size an arbitrary cardinal κ. As a consequence the structure of injective representations of noetherian κ-trees is completely determined. In the second part we will consider the problem whether arbitrary trees are source injective representation quivers or not.

We give a sufficient condition for the class of Gorenstein injective modules be precovering: if R is right noetherian and if the class of Gorenstein injective modules, 𝒢ℐ, is closed under filtrations, then 𝒢ℐ is precovering in R-Mod. The converse is also true when we assume that 𝒢ℐ is covering. We extend our results to the category of complexes. We prove that if the class of Gorenstein injective modules is closed under filtrations then the class of Gorenstein injective complexes is precovering in Ch(R). We also give a sufficient condition for the existence of Gorenstein injective covers. We prove that if the ring R is commutative noetherian and such that the character modules of Gorenstein injective modules are Gorenstein flat, then the class of Gorenstein injective complexes is covering. And we prove that over such rings every complex also has a Gorenstein injective envelope. In particular this is the case when the ring is commutative noetherian with a dualizing complex. The second part of the paper deals with Gorenstein projective and flat complexes. We prove that over commutative noetherian rings of finite Krull dimension every complex of R-modules has a special Gorenstein projective precover.

Given a double complex $X$ there are spectral sequences with the $E_2$ terms being either H$_I$ (H$_{II}(X))$ or H$_{II}($H$_I (X))$. But if $H_I(X)=H_{II}(X)=0$ both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with $X$. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article we give a different way of constructing homology groups of $X$ when H$_I(X)=$H$_{II}(X)=0$. With this result we give a new and elementary proof of balance of Tate homology and cohomology.

Let C be a set of modules. We argue that there is an ordinal κ such that if a module has a filtration by modules in C, then it has a filtration of length κ by direct sums of modules in C. As an application we give another way to prove a result of Saorín and Šťovíček and of Šťovíček.