2.4.4 Kernel of a ring homomorphism

Lemma 2..15   Let $\Phi$ be an affine ring homomorphism

$$\Phi : R=K[x_1,\ldots , x_m]/I\longrightarrow K[y_1,\ldots , y_n]/(g_1,\ldots , g_s)$$

given by $f_i=\Phi (x_i)\in K[y_1,\ldots , y_n]/(g_1,\ldots , g_s)\;$, $i=1,\ldots ,m\;$.
Then $Ker(\Phi )$ is generated by

$$(g_1(\underline{y} ),\ldots , g_s(\underline{y} )\,,\:(x_1-f_... ... )),\ldots , (x_m-f_m(\underline{y} )))\cap K[x_1,\ldots ,x_m]$$

in $K[x_1,\ldots , x_m]/I$ .

Remark 2..16   For $std(H) \cap R$ use lemma 2.7.

SINGULAR example:

ring r=...;
ideal null;
ideal F=...;
preimage(r,F,null);