Torsion-free Modules Research Articles

On finite-dimensional torsion-free modules and rings

1. This note is concerned primarily with a study of modules having zero singular submodule, called torsion-free modules, over finitedimensional rings. In some of our results we do not require that the ring be finite-dimensional but only that direct sums of torsion-free injective modules be injective, a property shown by Mark Teply to be equivalent to the ring containing no infinite direct sum of torsionfree left ideals. As might be expected, torsion-free modules over these latter rings behave very much like the usual torsion-free modules over (commutative) integral domains. For example, we show in Theorem 1 that direct sums of torsion-free injectives are injective if and only if every torsion-free module contains a unique maximal injective submodule. Also, in analogy to the case for Abelian groups, we establish that a torsion-free module over a finite-dimensional torsion-free ring contains a homomorphic image of every nonzero torsion-free module if and only if it contains a faithful injective submodule (Theorem 3). This enables us to show in Theorem 4 that a semiprime finite-dimensional torsion-free ring has a projective injective envelope if and only if it is semisimple Artinian. A result for torsion-free prime rings without any restricted chain conditions similar to Theorem 3 is also obtained. Finally, over self-injective finitedimensional rings we show that all torsion-free modules are injective and completely reducible (Corollary 6). We assume throughout that R is a ring with identity and all R-modules are unitary left R-modules.

ON HEREDITARY AND BASS ORDERS

A criterion is given for orders over Dedekind rings to be hereditary. It is proved that every finitely generated torsionfree module over a Bass order, i.e., an order of which every superring has injective dimension one, splits into a direct sum of ideals. A local description of Bass orders is presented.

Anti-isomorphisms of the endomorphism rings of torsion-free modules

Anti-isomorphisms of the endomorphism rings of torsion-free modules

On Simple Rings with Maximal Annihilator Right Ideals

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.We first prove the following:Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.

The Ring of Endomorphisms of a Torsion-Free Module

Journal of the London Mathematical SocietyVolume s1-39, Issue 1 p. 41-42 Notes and papers The Ring of Endomorphisms of a Torsion-Free Module E. H. Feller, E. H. Feller University of Wisconsin Milwaukee, Wisconsin, U.S.A.Search for more papers by this authorE. W. Swokowski, E. W. Swokowski Marquette University Milwaukee, Wisconsin, U.S.A.Search for more papers by this author E. H. Feller, E. H. Feller University of Wisconsin Milwaukee, Wisconsin, U.S.A.Search for more papers by this authorE. W. Swokowski, E. W. Swokowski Marquette University Milwaukee, Wisconsin, U.S.A.Search for more papers by this author First published: 1964 https://doi.org/10.1112/jlms/s1-39.1.41Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article.Citing Literature Volumes1-39, Issue11964Pages 41-42 RelatedInformation

A theorem on linear transformations of torsion-free modules of infinite rank

A theorem on linear transformations of torsion-free modules of infinite rank

Isomorphisms of the Endomorphism Rings of a Class of Torsion-Free Modules

Isomorphisms of the Endomorphism Rings of a Class of Torsion-Free Modules

Some Indecomposable Group Representations

In [1], Heller and Reiner have shown that any p-group G has infinitely many inequivalent, indecomposable rational integral representations, unless G is cyclic of order p or p2. In the present paper, we shall derive these results from a more general structural theorem in ? 2 below. This theorem states that, if 0 is any order in any algebraic number field K such that the p-group G has at least four distinct K-irreducible representations, then the group ring O[G] has infinitely many inequivalent, indecomposable torsion-free modules. From the available evidence, it seems that the converse to this statement ought to hold, i.e., that if G has three or fewer distinct K-irreducible representations, then O[G] has only finitely many distinct indecomposable torsion-free modules. Warning. We remark here once for all that any ring in this paper comes provided with an identity; any module is a left module; and the ring identity acts as the identity transformation of the module.

Torsion free covering modules

Let A be an integral domain and K its field of fractions. An Amodule E is said to be torsion free if ax = 0 for azA, x EE implies a = 0 or x = 0. We will say that a submodule E1 of an A -module E is pure in E if axE,=acEC\E1 for all aGA. Then if E is torsion free, a submodule E1 of E is pure in E if and only if E/E, is torsion free. Clearly the union of a chain of pure submodules of a module is still a pure submodule and if E2CE1, are submodules of E such that E2 is pure in E1 and E1/E2 pure in E/E2 then El is pure in E. It is well known that for any A-module E there exists a torsion free A -module E1 and an epimorphism p: E-*E1 such that if 4 is any linear mapping from E into a torsion free module F then there is a unique linear mapping f: El-*F such that f o p =4, i.e., the diagram

Torsion-Free and Divisible Modules Over Non-Integral-Domains

In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.

Isomorphisms of the endomorphism rings of a class of torsion-free modules

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

Torsion free and projective modules

Introduction. Serre has asked [5, p. 243] whether projective modules over the polynomial K[XI, * * *, X], K a field, are free, and Seshadri has proved that this is so when n =2. More precisely, he showed that every R[X]-projective module 'comes from one over R if R is either a principal ideal domain [6] or the coordinate of a nonsingular affine curve over an algebraically closed field [7]. By modifying Seshardri's argument in [6] we show here (Theorem 2.4) that R may be any Dedekind ring(2). Moreover we provide an example, due to S. Schanuel, which shows that some assumption like integral closure on R, is, in general, necessary for this conclusion. Problems of the above type can often be reduced to showing that a projective module is a direct sum of modules of rank 1, and this led us to ask for what integral domains R this is true for all (finitely generated) torsion free modules. Our main result (Theorem 1.7) gives an essentially complete solution to this problem. With mild assumptions on R, a necessary and sufficient condition is that every ideal in R can be generated by two elements. Thus these integral domains are very close to, but need not be, Dedekind rings. We assume a familiarity with the basic techniques of algebra as developed in Zariski and Samuel [9]. Homological algebra is appealed to in two discrete instances; one is a reference in the proof of Proposition 1.5, and the other is the proof of Lemma 1.9. The latter result, elementary as it is, seems to defy any proof which does not either use, or essentially reconstruct, the functor Ext'. Minor portions of this work were present in the author's doctoral dissertation. He gratefully records his indebtedness to Professor Kaplansky, who communicated a computation which suggested one of the essential ideas in the proof of Theorem 1.7. 1. Torsion free modules. Throughout this paper 'module means generated module, and ring means commutative ring. If M is anRmodule PuR(M) denotes the least number of elements required to generate M, and ,.t*(R) =sup PR (@) where 2t ranges over all finitely generated ideals of R. We call R local if it has a unique maximal ideal.

Isomorphisms of the endomorphism rings of torsion-free modules

THEOREM. For i = 1, 2 let Mi be a torsion-free module over the complete discrete valuation ring Ri, and suppose a is an isomorphism of E(R1, M1) upon E(R2, M2). Then the modules M1 and M2 are either both nondivisible or both divisible. In the first case, there exists a one-one semilinear mapping V of (R1, M1) onto (R2, M2) such that Sa = V-'SV for each S in E(R1, M1). In the latter case, Mi is a vector space over Ki (the quotient field of R,) and a is induced by a one-one semilinear transformation of (K1, M,) upon (K2, M2).

Decomposability of modules

It is an old theorem, essentially due to Prufer [5 ], that any torsionfree module of countable rank over a complete discrete valuation ring is a direct sum of modules of rank one. In [2] I generalized this to maximal valuation rings. In seeking to perfect this theorem, it is natural to ask: for what integral domains is it true that any torsionfree module of rank two is a direct sum of modules of rank one? There does not appear to be a simple answer. By changing the problem slightly, and confining the investigation to Noetherian domains, it is possible to single out the pleasant class of rings of the theorem below. However, the real reason for publishing this note is that the proof of Theorem 19 in [3 ] is, to put it politely, unconvincing.

Modules of Finite Rank Over Prufer Rings

In this paper the results of [4] are rederived and completed (in particular the question raised on p. 338 is completely settled) by a systematic use of extension theory [2, 3]. Instead of handling Dedekind rings and valuation rings separately, the treatment throughout is in terms of Prilfer rings; the modified descending chain condition in the one case and the hypothesis of almost maximality in the other are reformulated as linear precompactness (which serves also to exhibit their relationship to each other). The technique for obtaining the decomposition theorems is by and large the same: search for a pure submodule and proof that it is a direct summand. However, the concept of purity undergoes a closer analysis which permits one to disengage the relevant hypotheses. The rather successful definition of rank might also be mentioned. We shall be dealing exclusively with modules over commutative integral domains R for which the unit is the identity endomorphism. The most general rings we shall consider are the Prufer rings in which every finitely generated ideal is invertible: among these are the finitely principal domains (every two elements have a greatest common divisor) and,' more particularly, the valuation rings (in which this divisor is equal to one of the two elements). A module M is torsion-free if for every non-zero x E M, ax = 0 implies a = 0. A torsion-free module can be embedded in a vector space over Q, the quotient field of R; the dimension of the smallest subspace containing M (this is independent of the embedding) is called the rank of M. The rank of an arbitrary module M is defined as the smallest of the ranks of the torsion-free modules of which M is a quotient module. A module M is divisible if for every x E M and a E R (a $ 0) there exists y E M such that ay = x. This concept is in a way dual to that of torsion-free; in a divisible module the non-zero scalar multiplications are onto endomorphisms while in a torsion-free module they are one-to-one. The field Q (and more generally any vector space), as well as its quotient modules, are divisible; thus the rank of a module could also be defined as the smallest rank of a divisible module containing it.