If $R$ is a ring, $\mathfrak{a}$ is an ideal of $R$ and $S=A[x,y,z,\dots]$ where $A$ is a commutative ring, then is there a ring $S^\prime$ such that there is a bijection: $$\mathrm{Maps}(S,R/\mathfrak{a})\simeq\mathrm{Maps}(S^\prime,R)?$$ By maps I mean ring homomorphisms.
I know that if $\mathfrak{a}$ is the zero ideal then $S^\prime$ can be chosen to be $S$. But I have no idea on how to start when $\mathfrak{a}$ is a general ideal of $R$.