Power-series presentation of an adic completion ($R[[X_1,\dots ,X_s]]\twoheadrightarrow \hat{R}$)

If $I=(a_1,\dots ,a_s)$, evaluation $X_i\mapsto a_i$ gives a ring map to the $I$-adic completion, with each series evaluated modulo $I^r$ using only terms of total degree below $r$. It is surjective: successively represent the residual class in $I^r/I^{r+1}$ by a homogeneous degree-$r$ polynomial in the $a_i$. If $R$ is a commutative Noetherian ring, Noetherianity of formal power series rings therefore makes its completion Noetherian.