Torsion Submodule Research Articles

Strongly radical supplemented modules

Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.

Rad-Supplemented Modules

Let \tau be a radical for the category of left R -modules for a ring R . If M is a \tau -coatomic module, that is, if M has no nonzero \tau -torsion factor module, then \tau(M) is small in M . If V is a \tau -supplement in M , then the intersection of V and τ(M) is \tau(V) . In particular, if V is a \mathrm{Rad} -supplement in M , then the intersection of V and \mathrm{Rad}(M) is \mathrm{Rad}(V) . A module M is \tau -supplemented if and only if the factor module of M by P_\tau(M) is \tau -supplemented where P_\tau(M) is the sum of all \tau -torsion submodules of M . Every left R -module is \mathrm{Rad} -supplemented if and only if the direct sum of countably many copies of R is a \mathrm{Rad} -supplemented left R -module if and only if every reduced left R -module is supplemented if and only if R/P(R) is left perfect where R/P(R) is the sum of all left ideals I of R such that \mathrm{Rad}I = I . For a left duo ring R , R is a \mathrm{Rad} -supplemented left R -module if and only if R/P(R) is semiperfect. For a Dedekind domain R , an R -module M is \mathrm{Rad} -supplemented if and only if M/D is supplemented where D is the divisible part of M .

Jacobson Radical Isomorphism Theorems for Mixed Modules Part One: Determining the Torsion

Let R be a complete discrete valuation domain. A class of R-modules satisfies an isomorphism theorem if modules in that class are determined by their endomorphism rings. Do there exist classes of mixed R-modules for which the Jacobson radical of the endomorphism ring determines the structure of the modules? As a first step in answering this question, this article shows that there are two classes of mixed R-modules for which the Jacobson radical of the endomorphism ring determines the structure of the maximal torsion submodules of modules in that class.

Anticyclotomic Iwasawa theory of CM elliptic curves

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, the two variable main conjecture of Rubin implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule.

The splitting problem for coalgebras

Let R be a ring and let t be a torsion preradical, R is said to have the splitting property, provided that for every left R-module M, the torsion submodule t ( M ) of M is a direct summand of M. The characterization of rings with this property is a classical problem (in particular the Goldie and Dickson torsion theories have been studied) that for noncommutative rings remains open. We consider the problem for the algebra C * , associated to a coalgebra C, and the torsion preradical Rat. It is shown that if C * has the splitting property with respect a Rat, then C is finite dimensional.

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

Warfield modules and simply presented modules are considered whose torsion submodule is a direct sum of cyclics. A correspondence between Warfield modules and completely decomposable groups is established by showing that such a mixed module G is Warfield if and only if \(G/p^{\omega }G\) is simply presented and \(p^{\omega }G\) is completely decomposable. We give an example to show that this result cannot be extended to ordinals \(\sigma >\omega \). We also connect with the work of Nunke to characterize modules G whose zeroth Ulm factors are torsion in terms of \(G/p^\sigma G\) simply presented and \(p^\sigma G\) completely decomposable.

A remark on a theorem of Y. Kurata

In [K] Y.Kurata proved that the Goldie torsion theory splits centrally for dual rings. Here we extend his result to semilocal rings with left essential socle such that Soc (RR) ⊆ Soc (RR). An example will demonstrate that our observation extends Kurata’s result. All rings are associative ring with unit and all R-modules are unital. The singular submodule of a left R-module M is denoted by Z(RM). We abbreviate S := Soc (RR) and J := Jac (R) for the left socle resp. the Jacobson radical. We denote the left Goldie torsion theory, that is the hereditary torsion theory generated by all singular left R-modules, by τG and the torsion submodule of a module M by τG(M). τG is said to be centrally splitting if τG(R) is a ring direct summand of R. A classical result of Allin and Dickson [AD] states that τG is centrally splitting for a ring R if and only if R is a direct product of a semisimple ring and a ring with essential left singular ideal. Lemma 1. Let R be a ring with essential left socle S. Then (1) S = S ⊕ (S ∩ Z(RR)), where S is projective and R/S is τG-torsion. (2) J is τG-torsion if and only if S J = 0. Proof. The socle can be decomposed as S = S0 ⊕ S1 where S1 := S ∩ Z(RR) and S0 is a projective left R-module. S ⊆ S0, because for x, y ∈ S with x = x0+x1 and y = y0+y1 where x0, y0 ∈ S0 and x1, y1 ∈ S1. The product xy = xy0 ∈ S0 as xy1 ∈ SZ(RR) = 0. Thus S ⊆ S0 holds and there exists a left module S such that S0 = S ⊕ S. We have SS ⊆ S∩ S = 0. If RS is essential in RR, then S becomes singular (as it is annihilated by S) and must be zero as it is also projective. Thus S0 = S . Also R/S becomes τG-torsion as S1 ' S/S and R/S are 1991 Mathematics Subject Classification. 16S90.

Kähler differentials for points in [formula omitted

If R is the homogeneous coordinate ring of a reduced 0-dimensional subscheme of P n , we study the module of Kähler differentials Ω R/K 1 . Explicit presentations of it and its torsion submodule are used to describe the module structure. From this we derive many properties of the Hilbert function of Ω R/K 1 . Finally, this function is computed in a number of special cases, in particular for reduced 0-dimensional almost complete intersections.

Characterizing a class of simply presented modules by relation arrays

Simply presented modules and Warfield modules are described in a class of mixed modules \(\cal H \) with the property that the torsion submodule is a direct sum of cyclics and the quotient modulo the torsion submodule is divisible of arbitrary rank. Analogous to a result of Warfield it is shown that the mixed modules of torsion-free rank one are in some sense the building blocks of such modules. The results extend our previous work describing this class by relation arrays which are a natural outgrowth of their basic generating systems. Moreover, an intimate connection is shown between relation arrays and the indicators of modules. Furthermore, we prove two realizability results one of which is analogous to a theorem of Megibben for mixed modules of torsion-free rank one. One gives necessary and sufficient conditions on when a relation array of a module can realize an indicator of finite-type while the other shows that an admissable indicator can be realized by a simply presented module of torsion-free rank one.

Torsion in rank 1 Drinfeld modules and the uniform boundedness conjecture

It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We verify the conjecture for $r=1$, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of $K$. Moreover, we show that within an $\Lbar$-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over $L$ which have nonzero torsion. For the case $A=\Fq[T]$, $r=1$, and $L=\Fq(T)$, we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic $p$ function fields proves the uniform boundedness for the $\pp$-primary part of the torsion for rank~2 Drinfeld $\Fq[T]$-modules over a fixed function field.

The number of generators for the product of an ideal with its inverse

Let (R,m) be a commutative one-dimensional Noetherian local domain. Assume the integral closure satisfying asymptotic condition where b is a strictly positive C∞ function on the unit sphere in R2. is a discrete valvation ring and the residue fields of R and are the same. Let I be a non-principal ideal of R. A question, motivated by the work of Auslander in the early 1960's, is whether the torsion submodule of I ⊗R I−1 is always non-zero. In this paper we investigate the stronger condition, μR(I)μR(I−1) greater than sign μR(II −1)(where μ denotes the number of generators). We will show this inequality holds whenever the multiplicity of R is at most four.

Torsion in the tensor product of an ideal with its inverse

Let (R, m) be a one-dimensional Noetherian local domain. Assume the integral closure is a discrete valuation ring with maximal ideal . Let I be a non-principal ideal of R. A question, motivated by the work of Auslander in the early 1960's, is whether the torsion submodule of is always non-zero. In this paper we will show that if both m and I are generated by powers ofx, the residue fields of R and are the same and the multiplicity of R is at most seven, then . It follows that has torsion under these conditions.

Rigidity and Semi-invariants in Drinfeld Modules

We show that if φ is a Drinfeld A-module over a field L, then any polynomial g( x) ∈ L[ x] which maps the a-torsion into itself for all a ∈ A must be an endomorphism of φ. In generic characteristic, we prove a stronger result: if there is an infinite A-submodule S of the torsion submodule (over L̄) such that g maps S into S, or if g maps the a-torsion into itself for infinitely many a, then g is an endomorphism followed by a translation. The first of these results generalizes easily to maps between different Drinfeld modules. The second can be generalized as well, assuming a conjecture which would follow from an analogue of Serre′s theorem on the image of Galois. Analogous rigidity results are known to hold for abelian varieties. As one application, we show that the ring of semi-invariants of a Drinfeld module (defined by D. Goss) is nothing more than the ring of endomorphisms.

QF-3 endomorphism rings of Σ-quasi-projective modules

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

Divisible modules over discrete finite dimension valuation domains

In the sequel R denotes a discrete valuation domain (DVD) of finite Krull-dimension. This means that R is the ring associated with a valuation u with value group r= X;=, Z (lexicographically ordered) and n = dim (R). All modules considered are assumed to be unitary. It is well known that Dedekind rings can be characterized as those domains over which all divisible modules are injective [l, Proposition 5.11. Therefore there is a difference between divisible and injective R-modules if dim (R) > 1 and we can interpret the injective dimension as a measure of that difference. In this sense we will show that divisible and injective modules over finitely dimensional DVDs are as close as possible in terms of the above (Corollary 2.2). Finally we will study the structure of divisible R-modules and modules whose torsion submodule is divisible, making additional assumptions.

Torsion in the Cohomology of Line Bundles on Homogeneous Spaces for Chevalley Groups

In this paper we study the torsion submodules of the cohomology modules for line bundles on $G/B$ where $G$ is a Chevalley group over ${\mathbf {Z}}$ and $B$ a Borel subgroup.

Torsion in the cohomology of line bundles on homogeneous spaces for Chevalley groups

In this paper we study the torsion submodules of the cohomology modules for line bundles on G / B G/B where G G is a Chevalley group over Z {\mathbf {Z}} and B B a Borel subgroup.

Sur la ${\Bbb Z}_p$-torsion de certains modules galoisiens

Given an algebraic number field K and a prime number p let 𝒯 K be the Z p -torsion submodule of the Galois group of the maximal Abelian p-ramified p-extension of K. We want to study the Galois module structure of 𝒯 K . Under Leopoldt’s conjecture for K, 𝒯 K contains a submodule consisting of the semi-local p-primary roots of unity divided by the global ones, and the quotients of 𝒯 K by this submodule can be interpreted in two ways: either as the fixed points of an Iwasawa’s module, or as the Z p -torsion of a Bertrandias-Payan’s module. Applications are given to Iwasawa’s theory and to K-theory.

Alexander Modules

The Alexander modules of a link are the homology groups of the universal abelian cover of the complement of the link. For a link of $n$-spheres in ${S^{n + 2}}$, we show that, if $n \geqslant 2$, the Alexander modules ${A_2}, \ldots ,{A_n}$ and the torsion submodule of ${A_1}$ are all of type $L$. This leads to a characterization, below the middle dimension, of the polynomial invariants of the link. These results were previously proven for the special case of boundary links.

Alexander modules

The Alexander modules of a link are the homology groups of the universal abelian cover of the complement of the link. For a link of n n -spheres in S n + 2 {S^{n + 2}} , we show that, if n ⩾ 2 n \geqslant 2 , the Alexander modules A 2 , … , A n {A_2}, \ldots ,{A_n} and the torsion submodule of A 1 {A_1} are all of type L L . This leads to a characterization, below the middle dimension, of the polynomial invariants of the link. These results were previously proven for the special case of boundary links.