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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]?

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]?

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