From: "Charles H. Giffen" <chg4k@virginia.edu>
Subject: Re: Best text (Herstein?) for Abstract Algebra self-study?
Date: Mon, 10 Apr 2000 16:46:21 -0400
Newsgroups: sci.math
To: Pertti Lounesto <Pertti.Lounesto@hit.fi>
Summary: [missing]

Pertti Lounesto wrote:
> 
> D012560c wrote:
> 
> > > At Berkeley they use the text "A First Course in Abstract Algebra, 6th
> > > ed." by J. B. Fraleigh for the regular introductory Abstract Algebra
> > > course
> >
> > I took a correspondence course in Abstract Algebra about 5 years ago from the
> > University of Wisconsin at Madison and found this book very readable.  I like
> > this one better than Herstein's book.
> 
> J.B. Fraleigh: "A first course in abstract algebra", 5th ed.,
> contained an false theorem, on page 442, about division
> algebras.  Fraleigh defines division algebra as an algebra
> whose every non-zero element has an inverse, and then
> claims that such an algebra has dimension 1,2,4 or 8.
> Here is a counterexample, a 3D algebra, whose every
> non-zero element admits an inverse:  R^3 spanned by
> 1,i,j with multiplication rules i^2 = j^2 = -1, ij = ji = 0.
> 
> Is this false theorem still in the 6th edition of Fraleigh?

This seems to be a fairly common mistake.  It occurs in Birkhoff
& MacLane's "A survey of modern algebra" (at least in the first
edition on pp. 214-215) -- and also in MacLane-Birkhoff's
"Algebra" where they give the wrong definition of a field
(commutative ring in which every nonzero element has a 
multiplicative inverse).

These two mistakes also occur in Dieudonne's "Algebre lineaire 
et geometrie elementaire", Chap. I & App. IV.

In Jacobson's "Basic algebra I", the correct definition of a
division ring is given, but then he says that this is equivalent
to "1 =/= 0, and for any  a =/= 0  there exists  b  such that
ab = 1 = ba."  Of course, Jacobson also gives the (false)
exercise (S & T are sets):  Show that  a:S -> T is injective
if and only if there is a function  b:T -> S  such that
ba = 1_S -- which is impossible if  S  is empty and  T
is nonempty!

I guess even the famous mathematicians goof now and then.

--Chuck Giffen