Let $A=\mathbb{C}[X_1,X_2,X_3]/(X_1X_2X_3,X_1+X_2+X_3-1)$ and $B= \{ f(x_1,x_2,x_3) \in A \mid\forall \rho \in \mathfrak{S}_{3}, f(x_1,x_2,x_3)=f(x_{\rho(1)},x_{\rho(2)},x_{\rho(3)}) \}$ where $x_i$ denotes the image of $X_i$ in $A$.
Is it true that $B=\mathbb{C}[x_1+x_2+x_3,x_1x_2+x_1x_3+x_2x_3,x_1x_2x_3]=\mathbb{C}[x_1x_2+x_1x_3+x_2x_3]$?
I know that any symmetric polynomials is a polynomial in the elementary symmetric functions, but I don't know it is true for quotient of polynomial rings.