Zero element in an inverse limit in category of rings

Question

Let $(R,f_m)$ be the inverse limit of an inverse system $(R_m,f_{mn})$ of commutative rings with $1$. Let $x \in R$ be s.t. $f_{m}(x)=0$ for all $m$. Is it then true that $x$ must be the $0$ element of $R$? I tried to play with the universal property of $R$ but failed to figure it out, one way or another. Any hints?

Answer 1

This follows immediately from the construction of $R$ as a subring of the product $\prod_m R_m$. But you can also argue with the universal property: $x$ corresponds to a homomorphism $x : \mathbb{Z}[T] \to R$. Let $0 : \mathbb{Z}[T] \to R$ be defined by $T \mapsto 0$. By assumption, we have $f_m x = f_m 0$ for all $m$. Since $(f_m : R \to R_m)$ is a limit cone, in particular a mono-source, it follows $x = 0$.