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.