From: "Robin Chapman" Subject: Re: simplest proof for the AB=I => BA=I Date: Sat, 22 Sep 2001 07:56:44 +0000 (UTC) Newsgroups: sci.math Summary: When are left-inverses also right-inverses? "lssiu" wrote in message news:7c9c328b.0109210648.2610c7f2@posting.google.com... > The result, mentioned in the subject line, holds generally for all > division rings, even a larger subclass of rings. First come to my mind > is a proof that using the formula A adj(A)=adj(A) A=det(A)I, but it is > meaningful for commutative rings only. But I wonder if there exists > some simpler proof which holds for larger subclass of rings and does > not require the notion of det? If R is a left (or right) Noetherian ring, then ab = 1 for a, b in R implies ba = 1. If R is left Noetherian, then so is M_n(R), the ring of n by n matrices over R, and so AB = I implies BA = I in these rings. Division rings are trivially left Noetherian, so for these we certainly have AB = I implies BA = I. Robin Chapman www.maths.ex.ac.uk/~rjc/rjc.html -- Posted from webcacheh04a.cache.pol.co.uk [195.92.67.68] via Mailgate.ORG Server - http://www.Mailgate.ORG