We are dealing with modules of cotorsion pairs over commutative rings, especially when they are Σ -modules, i.e. their infinite direct sums also belong to the same class. We prove a few results about their structures and their direct decompositions. As an example, cotorsion pairs generated by finitely generated modules are briefly discussed.

Cellular covers which originate in homotopy theory are considered here for a very special class: divisible uniserial modules over valuation domains. This is a continuation of the study of cellular covers of divisible objects, but in order to obtain more substantial results, we are restricting our attention further to specific covers or to specific kernels. In particular, for h-divisible uniserial modules, we deal first with covers limited to divisible torsion-free modules (Section 3), and continue with the restriction to torsion standard uniserials (Sections 4-5). For divisible non-standard uniserial modules, only those cellular covers are investigated whose kernels are also divisible non-standard uniserials (Section 6). The results are specific enough to enable us to describe more accurately how to find all cellular covers obeying the chosen restrictions.

There are several characterizations of rings over which the modules admit certain covers (like injective, absolutely pure) or envelopes (like flat, torsion-free), not in the usual relation with cotorsion pairs. In this note we discuss commutative rings whose modules have divisible, h -divisible, or weak-injective covers, resp. commutative rings with weak-dimension 1 or projective dimension 1 preenvelopes. Subperfect rings and tight systems that are needed in the discussion are also dealt with in details.

We consider a generalization of a problem raised by P. Griffith [12] on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin [2]. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F ∼= T , then M is projective (Theorem 4.1). It is shown in Theorem 6.4 that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy [9]. The problem for valuation domains is also discussed, the results are similar to the case of abelian groups.

We prove a stronger form of an analogue of a Kaplansky lemma on homological dimensions by showing that in a pure-exact sequence 0 -> A -> B -> C -> 0, the weak dimensions of the modules satisfy w.d.B = max{w.d.A, w.d.C}. We also show that the same equality holds for the injective dimensions whenever the ring is noetherian. In addition, a version of Auslander's lemma for chains of pure submodules M-rho is proved: the weak dimension of the union of the chain equals the supremum of the weak dimensions of the factor modules M rho+1/M-rho in the chain. The same holds for injective dimensions if the ring is noetherian.

The aim of this note is to find those commutative rings over which an exact analogue of the structure theory of injective modules over commutative noetherian rings holds for weak-injective modules, i.e. for modules M satisfying $$\mathop {\mathrm{Ext}}\nolimits _R^1(A,M)=0$$ for all modules A of weak dimension $$\le 1$$ . We will show that, surprisingly, but a very few commutative rings R possess the property that their weak-injective modules admit (up to isomorphism) unique decompositions into direct sums of indecomposable modules each of which is the injective or the weak-injective envelope of a cyclic module of the form $$R/{\mathsf {p}} $$ with a prime ideal $${\mathsf {p}} $$ .

Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni-Salce [7] and Bazzoni [4], are generalized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs-Salce [14], showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost perfect rings within the class of commutative rings (Theorem 3.7). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair (P-1, D) (Theorem 4.1). Similar characterization is proved concerning the existence of divisible envelopes for h-local rings in the same class (Theorem 5.3). In addition, Bazzoni's characterization via direct sums of weak-injective modules [4] is extended to all commutative rings (Theorem 6.4). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.

In cotorsion theories, the cotorsion pairs (SF, MC) of strongly flat and Mattiscotorsion modules, and (F, EC) of flat and Enochs-cotorsion modules play important roles. We introduce a new cotorsion pair that in general lies properly between these two (in the partial order generally accepted for cotorsion pairs), and discuss its properties over commutative rings. In particular, we characterize the commutative rings over which this is a perfect cotorsion pair. Our results may shed more light on the relation between the two old cotorsion pairs.

We introduce a new class of commutative nonnoetherian rings, called n-subperfect rings, generalizing the almost perfect rings that have been studied recently by Fuchs and Salce. For an integer n >= 0, the ring R is said to be n-subperfect if every maximal regular sequence in R has length n and the total ring of quotients of R/I for any ideal I generated by a regular sequence is a perfect ring in the sense of Bass. We define an extended Cohen- Macaulay ring as a commutative ring R that has noetherian prime spectrum and each localization R-M at a maximal ideal M is ht(M)-subperfect. In the noetherian case, these are precisely the classical Cohen-Macaulay rings. Several relevant properties are proved reminiscent of those shared by Cohen-Macaulay rings.

The groups of equivalence classes of auto-equivalences of the categories of torsion and torsion-free modules over an integral domain R are determined. For the category of torsion-free modules, this group is isomorphic to the Picard group Pic R of R. We prove that for the category of torsion R-modules, this group is isomorphic to a group of certain divisible torsion groups called "cyclones" where addition is given in terms of the torsion product. In particular, this group is abelian and contains the direct product of the Picard group of the R-completion (R) over bar of R and of the group of "clones" of Q/R, where Q denotes the quotient field of R. The structure of the latter group is described in terms of the inverse system of the unit groups U(R/Rr) of the proper quotients R/Rr (r is an element of R) of R.

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings R/Rr are perfect rings (in the sense of Bass) for all non-zero-divisors r∈R. It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen–Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings $R$ with identity. The classical ring of quotients $Q$ of $R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums $\oplus Q$ one has to take the full subcategory of $Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which $Q$ has projective dimension $1$ (Theorem 6.4).

We continue our study of torsion groups concentrating on p-groups (with unspecified prime p) in the general case when the groups contain elements of infinite height. Matters are more subtle here as one has to deal with transfinite heights that are the central concept both in the search for invariants and in the proofs. The focus of the structure theory is on p-groups that can be described by their UK-invariants. Accordingly, this chapter is primarily devoted to countable p-groups and their generalizations: the totally projective p-groups. The theory is perhaps the most interesting and highly satisfactory classification of a fairly large class of p-groups in terms of well-ordered sequences of cardinal numbers (provided by their UK-invariants). The four main approaches to the theory of totally projective p-groups (simple presentation, total projectivity, nice systems, and balanced-projectivity) underline the extreme importance of these groups; this theory is unparalleled in beauty and richness in abelian group theory. Once the equivalence of the four main characterizations is established, there remain still some intriguing questions to be answered. For instance, which well-ordered sequences of cardinals may be the UK-invariants of a totally projective p-group? or, which is the largest class of p-groups that includes the generalized Prufer groups, is closed under direct sums and summands, and whose members are distinguishable via their UK-invariants? Needless to say, there have been various attempts to extend the well-rounded theory of totally projective p-groups, and various generalizations have been considered in the literature. So far these theories have produced only less remarkable results. Though several innovative techniques have been discovered, it seems that so far they have fallen short of true significance. The final sections of this chapter deal with questions that are spin-offs of the theory of totally projective p-groups, and offer a glimpse into classes depending on ordinal numbers.

This chapter continues the theme of torsion-free groups, this time for the infinite rank case. There is no shortage of relevant results.After a short discussion of direct decompositions of countable torsion-free groups, we enter the study of slender groups which display remarkable phenomena. We provide the main results on this class of groups. Much can be said about separable and vector groups. These seem theoretically close to completely decomposable groups, but are less tractable, and so more challenging. The measurable case is quite interesting.The theory of torsion-free groups would not be satisfactorily dealt with without the discussion of the Whitehead problem. For a quarter of century this was the main open problem in abelian groups. We will give a detailed proof of its undecidability, mimicking Shelah's epoch-making solution. We show that the answers are different in the constructible universe, and in a model of set theory with Martin's Axiom and the denial of CH.

In the preceding chapter we have encountered groups that were summands in every group containing them as pure subgroups: the pure-injective groups. In this chapter, we collect a large amount of additional information about these groups. Interestingly, these are precisely the summands of groups admitting a compact group topology, and the reduced ones are nothing else than the groups complete in the Z-adic topology. From Sect. 4 in Chapter 5 we know that every group can be embedded as a pure subgroup in a pure-injective (i.e., in an algebraically compact) group, and here we show that the significance of this embedding is enhanced by the fact that minimal embeddings exist and are unique up to isomorphism. Thus the theory of algebraically compact groups runs, in many respects, parallel to the theory of injective groups, a fact that was first pointed out by Maranda [1].The theory of algebraically compact groups is quite satisfactory: these groups admit complete characterization by cardinal invariants. We shall often meet algebraically compact groups in subsequent discussions.We close this chapter with the discussion of the exchange property. This is a remarkable, but rather rare phenomenon. Groups with this property show the best behavior as summands.

Most perfect objects in the category of abelian groups are those groups in which we can also 'divide:' for every element a and for every positive integer n, the equation nx = a has a (not necessarily unique) solution for x in the group. These objects are the divisible groups which are universal in the sense that every group can be embedded as a subgroup in a suitable divisible group. The divisible groups form one of the most important classes of abelian groups. In our presentation, we focus on their most prominent properties, many of them may serve as their characterization. Their outstanding feature is that they coincide with the injective groups, and as such they are direct summands in every group containing them as subgroups. Moreover, they constitute a class in which the groups admit a satisfactory characterization in terms of cardinal invariants. The concluding topic for this chapter is concerned with a remarkable duality between maximum and minimum conditions on subgroups.

In this chapter we start the discussion of torsion-free groups. First, we deal with general properties along with the finite rank case, and delegate the in-depth theory of torsion-free groups of infinite rank to the next chapter. After presenting the basic definitions and facts, we enter the study of balancedness, a stronger version of purity, which we have already met in the theory of torsion groups. Turning to the problem of direct decompositions, we start with the discussion of indecomposable groups; we do not restrict ourselves to the finite rank case as it seems more natural to deal with this important problem without rank restrictions. Concentrating on the finite rank case, the study of pathological direct decompositions is followed by positive results, the highlight being Lady’s theorem about the finiteness of non-isomorphic direct decompositions. Other aspects of direct decompositions are also discussed, including quasi- and near-homomorphisms. Finite rank dualities will also be dealt with.

The study of important classes of abelian groups begins in this chapter. Not counting the finite and finitely generated groups, the class of direct sums of cyclic groups is perhaps the best understood class.We give a fairly detailed account of free abelian groups, and discuss the presentation of groups via generators and defining relations. Several sections are devoted to direct sums of cyclic groups (called Sigma-cyclic groups); these groups share most useful properties, and can easily be characterized by cardinal invariants. We present a few criteria for such groups, and establish several remarkable results, e.g. Kulikov's theorem that passage to subgroups preserves Sigma-cyclicity. We draw attention to the method of smooth chains, which became the most important tool in the theory, and provides basic machinery for several results to come.We shall cover some of the aspects of almost free groups, but shall not pursue their theory farther, due to the sophisticated set-theoretical arguments required.In this chapter, in a number of proofs we have to use purity, so readers should be familiar with the fundamental results on pure subgroups (in Chapter 5) before studying the second part of this chapter.

In this chapter, we are going to discuss a basic concept: pure subgroup. This concept has been one of the most fertile notions in the theory since its inception in a paper by the pioneer H. Prufer. The relevance of purity in abelian group theory, and later in module theory, has tremendously grown with time. While abelian groups have been major motivation for a number of theorems in category theory, purity has served as a prototype for relative homological algebra, and has played a significant role in model theory as well.Pure subgroups, and their localized version: p-pure subgroups, are often used as a weakened notion of summands. In contrast to summands, most groups admit a sufficient supply of pure subgroups: every infinite set of elements embeds in a pure subgroup of the same cardinality. They are instrumental in several results that furnish us with criteria for a summand.Every group contains, for every prime p, a p-pure subgroup, called p-basic subgroup, that is (if not zero) a direct sum of infinite cyclic groups and cyclic p-groups. Basic subgroups are unique up to isomorphism, and store relevant information about the containing group. Basic subgroups were introduced by Kulikov for p-groups, and occupy a center stage in the theory of these groups.