Quotient Ring of a Polynomial Ring

[GargleBlast42]

Hi,

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

 

[Landau]

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

R/(I+J)\to (R/I)/q(J)

namely

r+I+J\mapsto q(r)+q(J)

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