Re: Rings, Restatement of earlier problem
From: Peter L. Montgomery (Peter-Lawrence.Montgomery_at_cwi.nl)
Date: 01/30/05
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Date: Sun, 30 Jan 2005 17:05:28 GMT
In article <cthg7m$erhh$1@netnews.upenn.edu>
"Tony" <Ttiger222@hotmail.com> writes:
>Find a ring with an element that has 2 left inverses.
Think about differentiation and integration of polynomials.
Given an antiderivative, perhaps 3*x^2 + 5*x - 7,
there is only one input giving this value, namely 6*x + 5.
But if we start with a derivative such as 6*x + 5,
there are many antiderivatives 3*x^2 + 5*x + C
with this derivative.
So derivative(some antiderivative(f)) = f for all f and
the derivative operator has multiple right inverses
"some antiderivative". You can modify this example to
get a ring with multiple left inverses.
>When I went to the errata of the author, I had found that the hint for using
>End(F_2 [x]) was changed.
>
>Now consider the ring End_k(V) where V is a vector space over k with basis
>{v_n : n >= 1}. Consider the map
>
>a : V ---> V
>
>that sends v_n to v_(n+1)
>
This is much like the antiderivative operator sending
X^n to X^(n+1)/(n+1).
>Then I have found one left inverse, that is :
>
>b : V ---> V
>
>that sends v_n to v_(n-1) and v_1 to v_1. Note that this is well defined as
>a left inverse, but not as a right inverse.
>
>I can't seem to think of anything else though...can anyone think of a
>different left inverse of a?
>
>Thanks for any help!
>
>Tony
>
>
-- Five years ago we feared Y2K disasters. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA
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