The Proper Class Generated by Weak Supplements

Abstract

We show that for hereditary rings the smallest proper classes containing respectively the classes of short exact sequences determined by small submodules submodules that have supplements and weak supplement submodules coincide. Moreover we show that this class can be obtained as a natural extension of the class determined by small submodules. We also study injective projective coinjective and coprojective objects of this class. We prove that it is coinjectively generated and its global dimension is at most 1. Finally we describe this class for Dedekind domains in terms of supplement submodules. © 2014 Copyright Taylor and Francis Group LLC. © 2013 Elsevier B.V. All rights reserved.