Torsion Classes Research Articles

Is the lattice of torsion classes algebraic?

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not. As a consequence of this, one obtains a (local) decomposition theorem which is canonical (in every sense of the word) and always works.

Torsion theories and ring extensions

Let ϕ:R→S be a ring homomorphism, and let ϕ∗:Mod-S -> Mod-R be the “restriction of scalars” functor. If Tis a hereditary torsion class for R, then T ∗={MεMod-S ≤ϕ∗(M)εTis a hereditary torsion class for S. We compare the quotient functors Qand Q∗of Tand T ∗respectively. Let ϕ∗:Mod-R→ Mod-S be the “extension of scalars ” functor:ϕ∗(M)=M⊗RS ∀MεMod-R. If ϕ∗ Q Q ∗is called S-good. If Tis S-good, then ϕ∗Q∗is a subfunctor of Qϕ∗If Sis also left R-flat, then ϕ∗Q∗≌,Qϕ ∗.We make an extensive investigation of the S-good property and give a number of ex-amples. The results are then applied to group rings and skew polynomial rings, where the invariance of Tunder ring auto-morphisms plays an essential role. In particular, for a group ring AG, if H◃ Gand Tis a hereditary torsion class for AH which is G-invariant, then Tis AG-good, and with equality if and only if G:H< ã or G:H=ãand Qcommutes with all free sums. In the special case when the right singular ideal of AHis zero and Tis the Lambek tor-sion class, the second condition above reduces to the condition that Q(AH)be semi-simple artinian.

Generalizations of the Simple Torsion Class and the Splitting Properties

In this paper all rings R are associative rings with identity and all modules are members of R-mod, the category of unital left R-modules, unless the contrary is specifically stated.

Products of torsion classes of lattice ordered groups

Products of torsion classes of lattice ordered groups

On finite left localizations

Of the several types of noncommutativ e quotient rings, finite left localizations have structure most like that of the original ring. This paper examines finite left localizations from two points of view: As rings of quotients with respect to hereditary torsion classes, and as endomorphism rings of finitely generated projective modules. In the first case, finite left localizations are shown to be the rings of quotients with respect to perfect TTF-classes. In the second, they are shown to be the double centralizers of finite projectors. The first characterization (Corollary 1.2) shows that finite left localizations may be realized as endomorphism rings of finitely generated projective idempotent ideals. It follows that every left localization of a right perfect ring is finite. Finally, the finite projectors of the second characterization (Corollary 2.2) are shown to have endomorphism rings whose quotient ring structure is very closely related to that of the original ring. For example, every ring of left quotients of the endomorphism ring is Morita equivalent to a ring of quotients of the original ring.

Conditions under which all preradical classes are perfect hereditary torsion classes

(1974). Conditions under which all preradical classes are perfect hereditary torsion classes. Communications in Algebra: Vol. 2, No. 4, pp. 365-376.

Halving the Milnor manifolds and some conjectures of Ray

The peculiar properties of the 2 j − 2 {2^j} - 2 dimensional generators of unitary bordism (the 2 2 -primary Milnor generators) are related to the 2 j − 3 {2^j} - 3 dimensional indecomposable torsion classes of Alexander. This result is then used to confirm a conjecture of Ray concerning the generalised homology spectral sequence for M S p ∗ ( M U ) MS{p_\ast }(MU) . Finally it is noted that Ray’s conjecture to the effect that all classes in M S p ∗ MS{p_\ast } are detectable by KO-characteristic numbers must fail.

Overrings of commutative rings. III. Normal pairs

A pair of integral domains (A, B) is a normal (resp., QR-) pair provided that A is a subring of B and all intermediate rings are normal in B (resp., rings of quotients of A). The special case of B the field of fractions of A (e.g., Prüfer domains and Dedekind domains with torsion class group) has been studied in detail. It is shown that any domain A possesses a unique overring B maximal with respect to forming a normal (resp., QR-) pair with A. An explicit description of this overring and all the intermediate rings in terms of localizations A is obtained, and further details are provided in the presence of a noetherian-like condition on A. In addition, the “overring” characterizations of Prüfer domains are extended to “intermediate ring” characterizations of normal pairs.

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Some closure properties for torsion classes of abelian groups

In an earlier paper (B. J. Gardner, Pacific J. Math., 33 (1970), 109-116) the torsion classes of abelian groups which are closed under pure subgroups were characterized, and §§ 3-6 of the present paper are devoted to generalizations of results appearing there. If & is a homomorphically closed class of objects in an abelian category, a subobject A of an object B is called ^-pure if it is a direct summand of every intermediate subobject X for which XIA e . (This terminology is due to C. P. Walker). In particular, ^ may be a torsion class. The following question is investigated: If and Ήf are torsion classes of abelian groups, when is closed under ^f-pure subgroups? Although ordinary purity is not ^/-purity for any torsion class %f, a torsion class ^7~~ is closed under pure subgroups if and only if it is closed under ^Hrpure subgroups, where ^l is the class of all torsion groups. In §5, for an arbitrary torsion theory (*g/, ^ ) a rank function (^/-rank) is defined for nonzero groups in ^ . It is shown that every torsion class closed under ^/-pure subgroups is determined by its intersection with ^ and the groups of ^/-rank 1 it contains. When ^ = , the groups with ^ rank 1 are the rational groups, so the earlier results for ordinary purity suggest that in general some refinement of the representation should he possible. A further special case of the general problem is also solved: Let X and Y be rational groups, T(X), T(Y) the smallest torsion classes containing them. If X is a subring of the rationale then T(X) is always closed under T(Γ)-pure subgroups; if not, the condition is satisfied if and only if X has a greater type than Y. § 7 is devoted to proving the following result: A torsion class is closed under countable direct products, i.e. direct products of countable sets of groups, if and only if it is determined by torsion-free groups.

Rings of quotients of endomorphism rings of projective modules

This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right iϋ-modiile. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left ϋί-modules determined by the projective module. The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right ϋί-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left ^-modules and hereditary torsion classes of left E-modules which contain the canonical torsion class determined by PR. If QR and Qs are rings of left quotients with respect to corresponding classes, then P($$RQR is a finitely generated projective right Q^-module with Qs as its Q#-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful .R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left ^-generator, and R is the Sendomorphism ring of SP When SP is a projective left Sgenerator, the maximal rings of left quotients of R and S are related in the manner considered above.

Q-Divisible Modules

Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].

Rings whose modules form few torsion classes

Characterizations are obtained of rings R such that the only torsion classes (respectively, hereditary torsion classes) of left unital R-modules are {0} and the class of all modules.

The torsion submodule splits off

A classical question for modules over an integral domain is, "When is the torsion submodule t(A) of a module A a direct summand of AT' A module is said to split when its torsion submodule is a direct summand. Kaplansky has shown [-14] that if R is a Dedekind domain, then every module whose torsion submodule is of bounded order splits. The converse of this result has been shown by Chase [5]. Results of [15] and [3] show that every finitely generated module splits if and only if R is a Priifer domain. Finally, if every R-module splits, then Rotman has shown [19] that R must be a field. Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Almost all of these are special cases of "torsion theories" in the sense of Dickson [6]. Moreover, most of these torsion theories are hereditary (i.e., the submodule of a torsion module is torsion); and hereditary torsion classes are classes of negligible modules associated with a topologizing and idempotent filter of left ideals in the sense of Gabriel [12] (also see [17]). Some recent papers ([4], [7], and [11]) have dealt with splitting results for specific hereditary torsion theories over certain commutative rings. The main purpose of this paper is to continue the investigation of the splitting properties of a torsion theory of modules over a commutative ring. Some characterizations for the splitting of modules, whose torsion submodules have bounded order, are obtained (see definition of bounded order below and Theorems 2.2 and 4.6). In particular, these results generalize the abovementioned theorems of Chase [5] and Kaplansky [14]. Our results show that the splitting of modules whose torsion submodules have bounded order frequently forces non-zero torsion modules to have non-zero socles. This increases our interest in the smallest hereditary torsion class 6e containing the simple modules. The class ow has previously been used in the study of commutative Noetherian rings and (left) perfect rings (e.g., see [7] and [10] and their references). For a commutative ring R, we show that 6e(A) is a summand of each R-module A if and only if non-zero R-modules have non-zero socles. This generalizes the main results of [7] and [11]; moreover, in case R is a Dedekind domain, our result coincides with the above result of Rotman. We also examine the properties of a splitting hereditary torsion theory of modules over a local ring R (unique maximal left ideal). We show that if R possesses a non-trivial torsion theory (~--, #-) such that every finitely generated module splits, then R is an integral domain, and (Y-, ~ is Goldie's torsion

TTF-classes over perfect rings

For a ring R with unit, let RM denote the category of unitary left R-modules. Following S.E. Dickson [3], a(non-empty) class of R-modules is a torsion class in rM if is closed under factors, extensions, and direct sums. If, in addition, is closed under submodules, then is said to be hereditary.

Quotient overrings of integral domains

If D is an integral domain with quotient field K, then by an overring of D we shall mean a ring D′ such that D ⊂ D′ ⊂ K. Gilmer and Ohm in [GO], Davis in [D] and Pendleton in [P] have studied the class of integral domains D having the property that each overring D′ of D is of the form Ds for some multiplicative system S of D. In [D] and [P] a domain with this property is called a Q-domain and in [GO] such a domain is said to have the QR-property. Slightly altering the terminology of [D] and [P], we shall say “QR-domain” instead of “Q-domain”. Noetherian QR-domains are precisely the Dedekind domains having torsion class group [GO; p. 97] or [D; p. 200]. Pendleton in [P; p. 500] classifies QR-domains as Prufer domains satisfying the additional condition that the radical of each finitely generated ideal is the radical of a principal ideal. It is pointed out in [P; p. 500] that this characterization of QR-domains still leaves unresolved the question of whether the class group of a QR-domain is necessarily a torsion group. We show in §1 that a QR-domain need not have torsion class group. Our construction is direct; however, the problem can be viewed in terms of the realization of certain ordered abelian groups as divisibility groups of Prufer domains, and we conclude §1 with a brief discussion of this approach to the problem.

Quasi-injective modules and stable torsion classes

Quasi-injective modules and stable torsion classes

On torsion classes of Abelian groups

On torsion classes of Abelian groups

Overrings of commutative rings. II. Integrally closed overrings

Introduction. All in this paper are commutative with identity element, and all subrings possess the identity element of the containing ring. An overring of an integral domain denotes here a subring of the quotient field containing the given ring. We assume the reader conversant with the elements of commutative (Noetherian) ring theory presented in the textbook of Zariski and Samuel [8], especially in those passages dealing with quotient rings, integral extensions, and the elementary theorems on valuation rings. Let that book serve as a general reference for the well-known definitions and results we introduce here with little or no comment. It is a fact that every overring of a Priifer ring (in the Noetherian case, ring) is also a Priufer ring, and, a fortiori, integrally closed. Further, for some Priifer (e.g., the ring of integers) all overrings are of quotients. Our purpose here is the study of with such properties, namely: P-domains, integral domains for which all overrings are integrally closed, and Q-domains, integral domains for which all overrings are of quotients. In ?1 we see that P-domains are indeed Priifer and that Q-domains are a special class of Pruifer (in the Noetherian case, exactly those with torsion class group). In ?2 and ?3 we consider P-rings and Q-rings in the context of with divisors of zero. ?3, which deals exclusively with Noetherian rings, is essentially a consideration of the notion Dedekind rings in the presence of zero-divisors. ?2 treats a special case of (sometimes) non-Noetherian rings. Since the results of ?2 and ?3 include as special cases many of the results of ?1, the reader is due an explanation of this uneconomical style of exposition. The excuse is twofold: (1) The consideration of integral domains contains the basic ideas of this study and serves to motivate the more general considerations. (2) The essentially simple ideas involved here tend to become obscured by the technicalities required in the presence of zero-divisors. We hope that this choice of exposition will produce a gain in clarity sufficient to compensate for the increased length. We shall attempt to avoid duplication of arguments by only