You would have to say what you mean by "equation satisfied by elements of $R$". But yes, there is a natural homomorphism $R \to R/I$, which by definition, preserves sums and products.
$R/I$ doesn't preserve EVERYTHING, though. For example, in $\Bbb Z$ we have:
$2\ast x = 0 \implies x = 0$, but this is NOT true in $\Bbb Z/(6)$ (we might have $x = 3$).