Re: Rings, Restatement of earlier problem

From: Bill Dubuque (wgd_at_nestle.csail.mit.edu)
Date: 02/05/05 

Date: 05 Feb 2005 12:10:55 -0500

Tony <Ttiger222@hotmail.com> wrote:
>
> Find a ring with an element that has 2 left inverses.

It's trivial generically: Z[w,x,y]/(wx-1,yx-1)

These elts are the right unit zero-divisors, i.e.

THEOREM The following are equivalent

1) x has more than one left inverse

2) x is a right divisor of 1 and 0

PROOF 1) => 2) yx = 1, wx = 1 => (y-w)x = 0

       2) => 1) yx = 1, zx = 0 => (y+z)x = 1

So equivalently generically: Z[w,x,z]/(wx-1,zx)

--Bill Dubuque

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