Re: Ring problem of the week

From: Todd Trimble (trimble1_at_optonline.net)
Date: 02/02/05 

Date: Wed, 2 Feb 2005 13:32:57 +0000 (UTC)

On 02 Feb 2005, Jose Capco wrote:
>Dear NG,
>
>I thought maybe this is true:
>
>If R is a ring and I,J are prime ideals of R with I contained in J, then
>there is an embedding of R/I in R/J ie. there is an isomorphism between
>R/I and a subring of R/J.
>
>I tried first finding the subring (assuming this were true) and I
>thought maybe S=R'\(J'*) works were R' and J' are the I residue of R and
>J and J'* is just J' without the zero. .. I found a mapping that would
>work as an isomorphism if I prove that S is closed under addition, but I
>wasn't yet able to do that (the mapping was in fact canonical). But is
>this true? If not, in which cases can one make R/I embed into R/J? Will
>appreciate any suggestions.
>
>Sincerely,
>Jose Capco

I think what you really want to say is that if I is contained
in J, then the canonical map R/I --> R/J is a *surjection*
(instead of injection).

Todd Trimble

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