Higher derivations on rings and modules

Paul E Bland

doi:10.1155/ijmms.2005.2373

Abstract

Letτbe a hereditary torsion theory onModRand suppose thatQτ:ModR→ModRis the localization functor. It is shown that for allR-modulesM, every higher derivation defined onMcan be extended uniquely to a higher derivation defined onQτ(M)if and only ifτis a higher differential torsion theory. It is also shown that ifτis a TTF theory andCτ:M→Mis the colocalization functor, then a higher derivation defined onMcan be lifted uniquely to a higher derivation defined onCτ(M).

Highlights

Rim has shown in [16] that under certain conditions a higher antiderivation d : M → M can be extended to a higher antiderivation dτ : Qτ(M) → Qτ(M), where Qτ : ModR → ModR is the localization functor [10] at a hereditary torsion theory τ on ModR
The purpose of this paper is to introduce higher differential torsion theories and to show that a higher derivation d : M → M can always be extended uniquely to Qτ(M), the module of quotients of M, if and only if τ is a higher differential torsion theory on ModR
We show that a higher derivation d : M → M can be lifted uniquely to the module of coquotients Cτ(M) of M at a TTF theory on ModR

Summary

Introduction

Rim has shown in [16] that under certain conditions a higher antiderivation d : M → M can be extended to a higher antiderivation dτ : Qτ(M) → Qτ(M), where Qτ : ModR → ModR is the localization functor [10] at a hereditary torsion theory τ on ModR. The purpose of this paper is to introduce higher differential torsion theories and to show that a higher derivation d : M → M can always be extended uniquely to Qτ(M), the module of quotients of M, if and only if τ is a higher differential torsion theory on ModR. We show that a higher derivation d : M → M can be lifted uniquely to the module of coquotients Cτ(M) of M at a TTF theory on ModR.

Differential torsion theory

Higher differential torsion theory

Higher derivations and modules of quotients

Higher derivations and modules of coquotients