Let $I$ be an ideal. Let $R= K[g_1 + I,\dots, g_n+I]$ be a sub-algebra of the affine K-algebra $A=K[y_1,\dots, y_m]/I$ for some $g_i\in K[y_1,\dots, y_m]$. Let $x_1,\dots, x_n$ be some aditional indeterminates, and let $J:= (I\cup \{g_1-x_1,\dots,g_n-x_n \})_{K[x_1,\dots, x_n,y_1,\dots, y_n]}$. I have show the subalgebra membership test. While doing this, I encountered some problems.
I have two questions:
a. if $h\in K[x_1,\dots,x_n]$ then why $h-h(g_1,\dots, g_n)\in J$?
b. Why if $f+I=h(g_1+I,\dots,g_n+I)$ then $f-h\in J$?