Quotient Ring of a Polynomial Ring

  • #1
Hi,

given a polynomial ring [tex]R=\mathbb{C}[x_1,\ldots,x_n][/tex] and an ideal [tex]I=\langle f_1, f_2 \rangle, \quad f_1, f_2 \in R[/tex], is it always true that [tex]R/I \cong (R/\langle f_1 \rangle)/\phi(\langle f_2 \rangle)[/tex], with [tex]\phi: R \rightarrow R/I[/tex] being the quotient map?
That is, is quotienting by I always the same as first quotienting by [tex]\langle f_1 \rangle[/tex] and then by [tex]\langle f_2 \rangle[/tex]?
 
Last edited:

Answers and Replies

  • #2
Landau
Science Advisor
905
0
Yes. If R is any ring with two ideals I, J, and q:R\to R/I is the quotient map, there is an obvious map

[tex]R/(I+J)\to (R/I)/q(J)[/tex]

namely

[tex]r+I+J\mapsto q(r)+q(J)[/tex]

It is easily shown to be a (well-defined) ring isomorphism.
 

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