Mon, March 28, 2022
4:00 pm - 5:00 pm
Zoom
Speaker: Mohamed Yousif (Ohio State University)
Title: Modules with the Exchange Property II
Abstract: A right R-module M is said to satisfy the (full) exchange property if for any two direct sum decompositions L=M⊕N=⊕i∈INi, there exist submodules Ki⊆Ni such that L=M⊕(⊕i∈IKi). If this holds only for |I|<∞, then M is said to satisfy the finite exchange property. A ring R for which RR has the finite exchange property is called an exchange ring. The exchange property is of importance because it provides a way to build isomorphic refinements of different direct sum decompositions, which is precisely what is needed to prove the famous Krull-Schmidt-Remak-Azumaya Theorem. It is an open question due to Crawley and Jonsson whether the finite exchange property always implies the full exchange property. In this talk we present the latest results on this open question and its relationship with the notions of perspective and clean rings and modules.