Let $R=R_1\oplus R_2$, where each $R_i$ is a commutative ring with unity. Let $(S,\eta)$ be a local subring of $R$. Let $\pi_1$ be the projection of $R$ onto $R_1$. It is also given that $\pi_1|_S$ is injective. Also all the rings involved here are Noetherian.
I need to show that if $\eta$ consists entirely of zero-divisors of $R$ then $\pi_1(\eta)$ consists entirely of zero-divisors of $R_1$.
Let $x=(x_1,x_2)\in\eta$ then $\exists$ $y=(y_1,y_2)\in R$ such that $xy=0$. But here $y_1$ can be $0$ also. I cannot proceed from this. Any help is highly appreciated.
This is used in theorem 6.4.5 in the book 'Cohen Macaulay rings' by Bruns and Herzog.